Mr. Trump seemed open to backing away from those sanctions in an interview with The New York Times in March, when he questioned whether anyone, other than the Obama administration, saw much use in them. His nominee for secretary of state, Rex W. Tillerson, has also been critical, not surprising because they have put a huge crimp in Exxon Mobil’s hopes for oil and gas exploration.

Syria could be the first area of cooperation. For months, Mr. Trump has talked about working hand in glove with Moscow against the Islamic State and other jihadist groups. That seemed a fanciful notion while Russia was painting all of the Syrian opposition with the same brush and bombing the moderate Syrian opposition more than the Islamic State. But if a shaky cease-fire agreement announced on Thursday holds, it could focus Russian military action for the first time exclusively on the Islamic State and the Islamist group Jabhat Fateh al-Sham, formerly known as the Nusra Front.

“The agreement potentially sets the table for Trump in Syria,” said Andrew J. Tabler, an expert on Syria at the Washington Institute for Near East Policy.

Still, Mr. Tabler and other Syria specialists said the opportunity came with a number of important caveats. First, the cease-fire needs to hold so that a more enduring political solution to the Syria crisis can be pursued. That will require restraint on the part of the government of President Bashar al-Assad, the Syrian opposition, the Iranians and the Russians.

Second, unless a political settlement is achieved that eventually eases Mr. Assad out of power, Syria may continue to be a magnet for extremists and insurgency, perpetuating the very problem that Mr. Putin and Mr. Trump say they are trying to solve.

While the Obama administration was not included in the cease-fire discussions, Russia’s foreign minister, Sergey V. Lavrov, extended a hand to Mr. Trump, who has never objected to Moscow’s growing influence in the Middle East.