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The anticipated dialogue takes place in the wake of a contentious NATO summit, and amid an ongoing investigation of Russia's attempts to influence the 2016 presidential campaign. What's more, Trump has insisted the two leaders meet at the beginning of the summit without any aides present — stirring concerns the former KGB officer will outflank his American counterpart.
"I think it's a good thing the president is talking to our adversaries, because that's what diplomacy is all about but certainly, we don't want him to go over there and give away the store," Michael Desch, a foreign policy expert and director of the International Security Center at the University of Notre Dame, told CNBC.
On Friday, some lawmakers pressed Trump to cancel the summit after Special Counsel Robert Mueller indicted 12 Russians for interfering in the general election. Yet on Friday, White House press secretary Sarah Huckabee Sanders told NBC News that the summit is "still on."
While it is unclear what Trump and Putin will (or won't) agree to next week, here's a summary of the issues they may discuss jointly.
What to look for: Trump may ask Putin for a new nuclear weapons agreement, as Russia accelerates its development of hypersonic and strategic weapons.
Background: In order to calm new fears of a budding arms race, Trump said he would discuss the reduction of nuclear weapons with Putin.
"If we can do something to substantially reduce them, I mean, ideally get rid of them, maybe that's a dream, but certainly it's a subject that I'll be bringing up with him," Trump said Friday. "The proliferation is a tremendous, I mean, to me, it's the biggest problem in the world, nuclear weapons, biggest problem in the world."
One option Trump may present to Putin is a new nuclear weapons agreement. The New START treaty, which is the current nuke agreement, is slated to expire in 2021.
"I think the low hanging fruit would be nuclear negotiations because the Europeans would be exceptionally happy to see us make progress with Russia on that front," explained Desch. "And so, reinvigorating the New START treaty would be a wise item on the agenda for the Trump administration."
Meanwhile, America's top nuclear commander has warned of Russia's sprint to deploy hypersonic weapons, a new breed of high-speed threats against which the U.S. is currently unable to defend.
"We don't have any defense that could deny the employment of such a weapon against us," Air Force Gen. , told the Senate Armed Services Committee in March. He added that both Russia and China are "aggressively pursuing hypersonic capabilities."
Hyten's comments came weeks after the Russian leader touted his nation's hypersonic weapons as "invincible" during a . Putin spoke in front of a projection showing video clips of the weapons as well as a simulated strike on the U.S. homeland.
"I want to tell all those who have fueled the arms race over the last 15 years, sought to win unilateral advantages over Russia, introduced unlawful sanctions aimed to contain our country's development: You have failed to contain Russia," Putin said during his address.
Of the six weapons Putin debuted in March, CNBC has learned that two of them will be ready for war by 2020, according to sources with direct knowledge of U.S. intelligence reports.
What to look for: Trump has been pressed to address Russia's meddling in the 2016 U.S. election in his conversation with Putin — but has been vague about it so far.
Background: Trump’s relationship with Putin has been the subject of endless controversy and speculation since he entered office, as federal investigations cast a shadow over the presidency.
Trump has vehemently denied all accusations of collusion, but he's also confounded allies and opponents alike with his frequent praise of the Russian strongman. Despite a consensus across America’s intelligence community that the Russians did indeed interfere in the 2016 election, Trump continually casts doubt on that assessment.
"Russia continues to say they had nothing to do with Meddling in our Election!" he wrote on Twitter on June 28.
Trump has said he would mention the topic of election meddling in his conversation with Putin. He did this once before during a meeting with the Russian president in Vietnam in November, appearing to take Putin at his word.
“He said he didn't meddle. He said he didn't meddle. I asked him again. You can only ask so many times, " Trump later told reporters.
Asked by reporters at a U.K. press conference on Friday if he'd press Putin on the issue again, Trump replied, "Meddling, we'll absolutely bring that up. I don't think we're going to have a 'gee I did it!' ... You never know what happens but I will absolutely, firmly ask the question."
Meanwhile, the Department of Justice on Friday named 12 Russian intelligence officers to be indicted for hacking into the Democratic National Committee's servers in 2016. That will likely serve as another point of contention for the two leaders on Monday.
What to watch for: Trump has suggested he may be flexible on U.S. policy concerning Crimea — and Putin will look to take advantage of this.
Background: Ukraine and Crimea have long dogged U.S.-Russia relations. The U.S. government refuses to recognize Russia’s 2014 annexation of Crimea, deemed illegal under international law.
But in recent weeks, Trump has left the door open for a reversal, saying only in response to questions on the topic: “We’ll see what happens.”
An about-face on Ukraine’s Crimean peninsula would completely upend U.S. foreign policy and its stated commitment to ally Ukraine.
Trump shocked officials at the G-7 meeting in June when he argued that the annexed Crimean peninsula should belong to Russia, because “people there speak Russian.” The assertion sharply contradicted longstanding U.S. and transatlantic policy of not recognizing the seizure of sovereign territory by force.
Since 2014, Washington has slapped sanctions on Russia for its invasion, and the ongoing war in Ukraine’s east, which has killed well over 10,000 people.
“Kiev’s main concern is that President Trump will unilaterally recognize Russia’s annexation of Crimea” — effectively selling it out to the Kremlin, said Daragh McDowell, senior Russia analyst at Verisk Maplecroft.
The legality of such a move — and whether it would mean a formal recognition of Russian sovereignty over the peninsula — is unclear, McDowell said. However, it would certainly demoralize U.S. allies and trigger domestic instability in Ukraine.
What to look for: Putin will undoubtedly raise the issue of sanctions relief, something he has sought since 2014.
Background: Moscow has been under U.S. and European sanctions since its 2014 illegal annexation of Ukraine's Crimean peninsula, which played a role in Russia's subsequent recession. But the U.S. government has followed this with a raft of new sanctions, in 2017 and 2018, despite Trump's apparent reluctance.
In August 2017, Trump signed into law the Countering America’s Adversaries Through Sanctions Act (CAATSA), a bill overwhelmingly passed by Congress. The president called it "seriously flawed" even as he signed it.
The sanctions applied to Russia, Iran and North Korea, and was designed to expand and codify punitive measures previously imposed by executive orders. It applied to a range of areas that included cyber security, financial institutions, human rights abuses, and transactions with Russian defense or intelligence sectors.
In March of this year, the U.S. government sanctioned Russian entities involved in cyber hacking during the 2016 election, and in April delivered a painful blow to Russia's elite: It sanctioned seven Russian oligarchs and 17 Russian government officials, as well as 12 entities owned by those oligarchs. The measures came in response to Russia's "malign" activities abroad, including human rights abuses, its military campaign in Syria, and political interference in other countries.
But as hard as Putin may try for sanctions relief, removing them is no longer in Trump's power since their codification into law under CAATSA, said Christopher Granville, Russia expert and director at TS Lombard.
"However, Trump does have the power not to impose any new ones," Granville noted. The American president has long been suspected of wanting to lift the sanctions, citing the need for improved relations with Moscow, and has deliberately missed previous congressional deadlines to impose them.
"That is certainly valuable in itself for the Russian economy, since what causes damage is not so much particular sanctions measures in themselves, but lack of visibility of what new sanctions escalations might be on their way," Granville said.
What to watch for: Putin may try to make a deal with Trump that would largely benefit Syrian President Bashar Assad and Iran.
Background: The last time the Trump administration crossed paths with Russia in Syria was in April, when the U.S. and its allies conducted precision missile strikes against the Assad regime. The strikes were in retaliation for a chemical attack carried out by Assad-supporters on Syrian civilians.
In a speech from the White House, Trump directly called out Russia and Iran, which back the regime of Assad.
"To Iran and to Russia, I ask: What kind of a nation wants to be associated with the mass murder of innocent men, women and children?" Trump said. "Hopefully someday we will get along with Russia, and maybe even Iran. But maybe not."
The missile strikes marked a dramatic reversal for Trump, who has frequently said he wants . His desires contradict U.S. military and national security advisers who see a more long-term role in Syria.
Meanwhile, the brutal conflict in Syria has largely been defined by foreign interventions.
Russia, Iran and Turkey are the three major powers poised to further influence the war-torn country, if the U.S. decides to pull out. Russia is Assad's strongest backer, alongside Iran.
Brett McGurk, the U.S. special envoy for the global coalition against ISIS, told a forum at the U.S. Institute of Peace in April that the U.S. mission in Syria was far from over.
Trump may push for some sort of bargain where Russia pulls its troops from Syria or ceases its support for Iran — perhaps in exchange for U.S. concessions on sanctions or recognition of Moscow's Crimea annexation. Still, Washington does not necessarily have the right under international law to make such offers.
What to watch for: In the wake of the Singapore summit, the first face-to-face meeting between a leader from North Korea and the U.S., Trump could ask Putin to leverage Russia's economic influence over Pyongyang.
Background: Trump may ask for Putin's help amid diplomacy discussions with the rogue regime, according to Benjamin Katzeff Silberstein, an associate scholar at the Philadelphia-based Foreign Policy Research Institute. Silberstein explained that Trump could ask the Russian leader to maintain sanctions on Kim Jong Un's regime.
Putin imposed restrictions on North Korea to comply with a U.N. Security Council resolution last October, but Moscow has generally rejected most U.S.-led efforts to isolate Pyongyang.
Russia's economic influence over North Korea is nowhere as great as China's, but it still wields considerable sway thanks to deep-rooted trade, cultural and commercial relations. The Eurasian country is a major destination for North Korean laborers and in 2014, it wrote off 90 percent of Pyongyang's $11 billion Soviet-era debt. In May, Kim said he valued Putin "highly" for opposing the U.S., according to Russian media.
But Putin won't be doing Trump any favors: Russia actually stands to benefit from a denuclearized North Korea.
"Russia is hoping that a potential cooling of tensions in the area will give it economic and geopolitical advantages," Silberstein told CNBC.
Meanwhile, Secretary of State Mike Pompeo, who is currently leading the negotiations with North Korea, will meet with Russian Foreign Minister Sergei Lavrov at the Helsinki summit next week.
CNBC's Nyshka Chandran in Singapore contributed to this report.