It was Mr. Assad’s second known trip abroad since the civil war in his country began in 2011; he visited Moscow in 2015, soon after Russia began its pivotal air campaign in support of his forces. The Sochi stop came as his other main allies, Iran and the Lebanese militia Hezbollah, declared that they had defeated the Islamic State in its last major Syrian stronghold, Bukamal, near the Iraqi border.
Saying that “the military operation is really coming to an end,” Mr. Putin told Mr. Assad that it was time to work toward a lasting political settlement.
Parallel Russian- and American-led campaigns against the Islamic State have largely shattered the group’s territorial self-declared caliphate. But a solution to the underlying conflict — which began after Mr. Assad’s forces cracked down on political protests — has remained elusive.
Insurgents unaffiliated with the Islamic State still hold patches of territory, besieged and bombarded by government forces, near the Syrian capital, Damascus; in the northern provinces of Aleppo and Idlib; and along the border with Jordan. Tens of thousands of people are missing, including civilians believed detained by the Syrian government. Twelve million Syrians, half the population, have been driven from their homes.
On Tuesday, the Syrian Network for Human Rights, which tracks war casualties through a database of victims identified by name, said that since March 2011, at least 26,446 children had been killed in the conflict, a vast majority by government forces.
“The main thing is to move to political processes,” Mr. Putin told Mr. Assad, according to Russian state news media. “I am pleased to see your readiness to work with everyone willing to establish peace and find solutions.”
Mr. Assad, who embraced the Russian leader and shook hands with a row of generals, replied, “We must admit that the operation made it possible to advance the process of political settlement in Syria.”
The United States and other international opponents have largely backed off their longstanding demand that Mr. Assad step down and have signaled willingness to accept a political transition that left him in power for at least some amount of time. But that remains unacceptable to many rebels and political opposition groups, and Mr. Assad has been accused in European courts of presiding over large-scale war crimes.
Suhair Atassi, an opposition activist, said the West appeared to be more and more willing to go along with Russia’s agenda for Syria.
Ms. Atassi was one of eight members to announce their resignation this week from the opposition negotiating group that has been attending the long-stalled talks sponsored by the United Nations in Geneva, including the committee’s leader, Riad Hijab. She said they were under pressure to make compromises they could not accept. .
“They told us very clearly, ‘You, the opposition, will be out of the political scene if you don’t accept the fact that Bashar is staying,’ ” Ms. Atassi said.
“For the first time, I feel the international community is united,” she added “They told us, ‘You can beat Bashar in the election.’ ”
Ms. Atassi said the main international debate over Syria’s future was now about “ small details — whether Bashar will run for the next elections or not, how long he will stay in his term, his powers.”
She said Western countries had become more and more inclined to simply hand over the thorny matter of Syria to Russia and accept Moscow’s approach.
The Trump administration official, who spoke on the condition of anonymity to describe the confidential call with Mr. Putin, said Mr. Trump’s emphasis on stability did not signal a change in American policy, just that the president considered peace the priority. Nor did Mr. Trump bring up Russia’s veto last week of a Security Council resolution that would have extended the investigation into who used chemical weapons in Syria, the official said.
“Things are obviously complicated with Russia,” a State Department spokeswoman, Heather Nauert, told reporters.
Noting that the administration believes that Mr. Assad’s military used chemical weapons multiple times against civilians, Ms. Nauert gamely attempted to square that with a photo from Monday’s meeting of Mr. Putin enveloping Mr. Assad in a hug.
“I think that hug is proof that Vladimir Putin bears a certain responsibility for trying to help out Syria,” she said, “for trying to get Bashar al-Assad to — well, trying to get that government — to put pressure on Bashar al-Assad so they don’t do something like that again.”
Opposition members like Ms. Atassi say lasting peace will be unattainable without accountability for war crimes like chemical weapons use and torture.
Russian state media said the Kremlin supported efforts by Saudi Arabia, which is the host for the opposition committee, to reorganize the opposition, presumably an effort to recruit a committee that would agree to new parameters. A new Geneva round is scheduled to begin next Tuesday.
Mr. Putin then plans to be the host at what he calls a Syrian Peoples’ Congress in Sochi on Dec. 2. Initial plans for the meeting called for inviting 33 Syrian opposition groups, some tolerated by Damascus, as well as Kurdish groups — a much broader range than those included in the Geneva process.
The meeting was postponed once over Turkey’s objections to inviting Kurdish groups that the Turks see as allied with Kurdish separatists in Turkey.
Another problem that remains unresolved is the fate of areas taken from the Islamic State by the Syrian Democratic Forces, a Kurdish-led, American-backed militia. The Syrian government has said it will fight to take the territory.
But the biggest problem for the Sochi talks, Neil Hauer, an analyst specializing in Russia’s involvement in Syria, wrote Monday, is “the opposition’s deep mistrust for Russia as a serious broker for an inter-Syrian settlement.”
Even before calling for the Sochi meeting, Russia had already carved out a parallel process to Geneva — talks in Astana, Kazakhstan, it has led with Iran and Turkey. That process, Mr. Hauer wrote, has largely functioned as an effort by those countries “to impose their own policies on Syria regardless of the whims of local actors.”
But some Syrian rebels have taken part in the talks, saying they see little option. The main result of the Astana talks has been the creation of four so-called de-escalation zones, which were meant to calm violence but where in practice heavy bombardment continues, hitting trapped civilians as well as armed groups.
Also on Tuesday, President Hassan Rouhani of Iran declared “the end” of the Islamic State and congratulated the peoples of Syria, Iraq and Lebanon as well as “dear brother” Qassem Soleimani, the commander of the Islamic Revolutionary Guards Corps’ elite Quds Force, who played a major role in battles in Iraq and Syria.
His remarks echoed those made on Monday by Hassan Nasrallah, the leader of Hezbollah, the Iran-backed Shiite militia group. Mr. Nasrallah spoke in more detail than he had previously done about Hezbollah’s involvement in Iraq, saying the group had sent many commanders to Iraq to aid in the battle against Islamic State.
Hezbollah has played an even larger and more prominent role in Syria. The new power, influence and weapons it gained in the process have roiled the region, as Saudi Arabia takes increasingly assertive steps to push back.
Mr. Nasrallah said his fighters would leave Iraq if they were no longer needed.
But as the United States has learned repeatedly, declaring “mission accomplished” in Iraq does not mean the fight is over.