Political analysts say Saudi Arabia's King Salman and Crown Prince Mohammed bin Salman may scale back the war in Yemen and the blockade of Qatar. Such moves would allow Saudi Arabia "to show some good faith to the U.S. and to retain the high level of strategic support that it has from the White House," Emily Hawthorne, Middle East and North Africa analyst at think tank Stratfor, told CNBC.
Saudi officials did not respond to a CNBC inquiry about whether they would change their Yemen or Qatar policies.
In April, Secretary of State Mike Pompeo told officials in Riyadh to end its land, maritime and air blockade against Qatar. Saudi Arabia — along with Bahrain, the United Arab Emirates and Egypt — severed ties with Qatar last year, accusing it of supporting terrorism. Qatar denies that it supports terrorism.
Last month, the U.S. increased its calls for a cease-fire in Yemen, where Houthi rebels, backed by Iran, have been fighting a Saudi-led coalition that supports the Yemeni government.
In a lengthy statement on Tuesday, Trump said the United States stands with Saudi Arabia in the wake of journalist Khashoggi's death. The Washington Post columnist and critic of the Saudi royal family was killed at the Saudi consulate in Istanbul last month.
The CIA has determined that the Saudi crown prince ordered Khashoggi's assassination, according to Friday reports from NBC News and the Washington Post. Saudi Arabia rejects those claims.
Trump, on Tuesday, said the CIA had "nothing definitive" on the crown prince's involvement. He claimed that a "very full report" about the U.S. investigation would arrive, though he provided no timeframe for such a report.
The only punitive measure Washington has taken is the sanctioning of 17 Saudi officials for their role in the killing.
Faced with an international uproar over its alleged role in Khashoggi's death, Riyadh will now take care to protect its relations with the West, according to Ayham Kamel, head of Eurasia Group's Middle East and North Africa research team.
"The U.S. administration for months has been critical of the dispute among the Gulf allies, but it has been unable to incentivize Saudi Arabia to shift its stance," the analyst wrote in a recent note.
Saudi leadership "is not interested in adopting any policies that could cause relations with the West to implode," so Riyadh is now likely to ease tensions with Doha and de-escalate the Yemen war in the coming months, Kamel continued.
Speaking to CNBC on Wednesday, Saudi Arabia's Foreign Minister Adel al-Jubeir said his country seeks an end to the hostilities in Yemen and expressed hope for Houthi forces to negotiate a solution at upcoming peace talks in Sweden.
"The political solution was available from day one, but the Houthis refused to take it," al-Jubeir told CNBC's Hadley Gamble on Wednesday.
"Saudi Arabia will continue its foreign policy of trying to bring peace and stability to the region, and trying to push back against extremism and terrorism, against Iran's nefarious and malign and aggressive policies," al-Jubeir said.
He also rejected reports suggesting a change in the kingdom's line of succession.
Regarding Khashoggi, al-Jubeir protested that Riyadh was declared guilty by people without a full assessment of the facts.
Richard Murphy, formerly U.S. ambassador to Saudi Arabia and U.S. assistant secretary of state for Near Eastern and South Asian Affairs, told CNBC that Trump should take advantage of his new sway over Riyadh to make foreign policy gains.
For instance, Murphy said, Trump could pressure Riyadh to get a cease-fire in Yemen while withholding American arms and intelligence assistance for the fighting there.
Regarding Trump's pick for the next U.S. ambassador to Saudi Arabia, John Abizaid, Murphy expressed confidence in Abizaid's ability to deal with the kingdom at such a sensitive time.
The United States has long relied on Riyadh to counter Iran in the Middle East and to combat terrorism in the region.
It has also counted on Saudi Arabia to restrain oil prices and to purchase U.S.-made weapons. Saudi Arabia is OPEC's biggest oil producer and the top buyer of U.S. weaponry.
The United States and Saudi Arabia have dealt with serious obstacles in their history, such as the fact that most of the hijackers in the Sept. 11, 2001 terror attacks were Saudis. Even then, Washington imposed no sanctions on Saudi officials.
The Khashoggi crisis is "probably the toughest issue that they're going to have to get past," Hawthorne said. "The White House isn't looking for partners that are the pinnacle of human rights and press freedom ... they are looking for partners that can help them achieve other strategic objectives."