On the whole, Turzanski said Trump probably views more cooperation in the terror fight as "a good starting point" to begin to improve the U.S.-Russia relationship. For Putin, working together with the new U.S. administration on priorities such as global terrorism and Islamist fundamentalists "won't put him in a position where it's going to cost him a lot."
One person likely to be critical in any negotiation is ExxonMobil CEO Rex Tillerson, Trump's choice for U.S. secretary of state and a longtime friend of Putin. Tillerson is known to have negotiated commercial deals with world leaders, and if confirmed by the Senate, will be asked to put his skills to use as America's top diplomat.
Others are not so certain the Russians want to take on ISIS, which now poses threats beyond the Middle East with members or its terrorist sympathizers.
"In reality, my view is that the Russians aren't all that interested in defeating ISIL," said Eugene Rumer, director of the Carnegie Endowment for International Peace's Russia and Eurasia program in Washington. "It's not as much of a priority to them as it is to us and our European allies."
Rumer, a former national intelligence officer for Russia and Eurasia at the U.S. National Intelligence Council, said the priority for Moscow has been keeping the Assad government in power. Recent ceasefires with opposition forces fighting Assad could form the basis of an eventual political solution for Syria. The civil war in Syria has contributed to a refugee crisis and claimed at least 400,000 lives, according to the United Nations.
"You hear a lot of talk about a 'grand bargain' — a comprehensive deal," Rumer said. "Now that they've achieved a breakthrough in Syria, they will offer something in exchange there. They will not give up Assad and they … will maintain a military and naval facility in Syria."
It's unclear how any deal with the Russians over Syria might look in a Trump administration, but experts suggest it could involve a higher level of cooperation in fighting global terrorism. Also, they suggest Putin may be motivated to give "an election present" to Trump to score points for future issues.
At the same time, better ties with Putin might help the new U.S. administration offset Chinese ambitions, which include the Beijing government's island buildup in the South China Sea and other military and economic actions.
China's defense budget — already second to the U.S. in terms of annual spending — is on track to almost double in the 2010 to 2020 period and overtake the entire Western European regional defense expenditures, according to the annual Jane's Defence Budgets Report released this month by IHS Markit. Meanwhile, it predicts Russia's defense spending will decline this year for the first time since the late 1990s.
Some analysts argue that the history of personal animosity between Obama and Putin probably doomed the chances of a thaw in icy relations between the U.S. and the Kremlin. They also say any deal Trump gets that might ease economic sanctions on Russia will still face opposition from Russian hawks in the Senate.
Then again, there are international policy experts skeptical Trump can turn things quickly with Moscow. The Iran deal made by the Obama administration, which included lifting international oil and economic sanctions against Tehran, was criticized by Trump during his campaign. Rescinding the nuclear deal could raise alarm in Russia and would also impact other countries that are part of the agreement.
"If the Trump administration attempts to reset with Russia, it will be the fourth administration in a row to try such a reset at the start," said Christopher Chivvis, associate director of the RAND's International Security and Defense Policy Center.
Chivvis, who worked as a Eurasian security policy officer in the Office of the Secretary of Defense during the first term of the Obama administration, added: "Trump can try it too. But the reality at least in my analysis is that the U.S. and Russian interests are not aligned on most issues."