"You can't just ask Russian Federation to redefine its interests", he said, adding that "Putin doesn't do favors".
Activists including the Britain-based Syrian Observatory for Human Rights said buses carrying rebels began leaving the rebel-held towns Madaya and Zabadani near Damascus on Friday morning in the first phase of the deal, which will also see the evacuation of residents from two pro-government Shiite villages in northern Syria.
But neither side cited much in the way of concrete new avenues of cooperation, beyond establishing a working group to address what Tillerson called "smaller issues".
Russian Foreign Minister Sergey Lavrov, who hosted his Iranian and Syrian counterparts at Friday's trilateral meeting in Moscow, denounced last week's US attack on Syria and warned that any further such action would entail "grave consequences not only for regional but global security".
The military action, coupled with American officials' change of tone over Assad's future, stirred speculation that the U.S. and its Western allies were ready to put boots on the ground in Syria.
But that prospect appeared remote, with Trump's secretary of state Rex Tillerson getting a frosty reception in Moscow, and Russian Federation using its UN Security Council veto to swat down a US-backed resolution demanding Syrian cooperation in probing last week's suspected chemical attack.
"I felt we had to do something about it".
"I think he fundamentally doesn't understand the nature of the Russia-China relationship", she said, describing it as pragmatic.
Tensions between Moscow and Washington have further aggravated over Syria after the U.S. missile strike on a Syrian military airfield on Thursday, which it claims to be in response to the alleged Syrian government's chemical weapons attack last Tuesday in Khan Sheikhoun. On the campaign trail, Trump pledged to improve relations with Russia - only to see his plans torpedoed by a combination of suspicions over Russian interference in the US election on Trump's behalf, and speedy clashes between the USA and Russia on the issue of Syria and the fate of Assad.
The toxic gas attack on April 4 prompted the United States to launch missile strikes on a Syrian air base and widened a rift between the United States and Russian Federation.More news: The tone of tensions: Korean voices, North and South
Earlier Russia's President Vladimir Putin rejected allegations that Syria was behind the chemical attack, saying Syria had given up its chemical stockpile. Tillerson also broke with his predecessors' tradition of meeting members of Russia's civil society; it wasn't clear whether the snub was intended as an overture to Putin, or a genuine lack of interest by the Trump administration.
USA president Donald Trump's White House victory had raised hopes in Moscow that some kind of rapprochement was possible between the former Cold War foes. "We will not accept that, we don't believe the world will accept that", Tillerson said.
Tillerson meets Putin amid "low point" in US-Russia tiesThe U.S. said relations with Russia are at a 'low point, ' but both countries said they are willing to improve ties. "That could happen, and it may not happen, It may be just the opposite", Trump said.
Iran, Russia and Syria deny that the authorities in Damascus were behind the attack, in which more than 80 people were killed and hundreds injured. The visit came amid a swift and severe deterioration in U.S.
A quick détente seemed a remote possibility, given the level of tension surrounding the aftermath of the Syrian chemical weapons attack.
I support President Trump's military response to Assad's latest use of chemical weapons.
But Russian foreign minister Sergey Lavrov responded that "not a single fact has been confirmed". -Russian relations with the state television channel Mir before his meeting with Tillerson.
"Every administration tries to improve relations, but there is a very basic fundamental fact: Across the geopolitical chessboard the US and Russian Federation have fundamentally very different interests", said Harry Kazianis, a senior fellow for defense policy at the Center for the National Interest, a Washington think tank advocating "realistic" foreign policy.
Russian Federation said the US strike was illegal though and Lavrov repeated Moscow's stance on Wednesday, saying an global investigation should be left to determine who was to blame and what happened. The agreement was suspended after the USA launched its first attack on Syria during the country's six-year conflict last week. "We're going to see how that all works out".