"The Secretary looks forward to further meetings with the Foreign Minister to discuss the respective roles of the United States and Russia in de-escalating the conflict and supporting the talks in Geneva to move the political solution forward," State Department Spokesperson Heather Nauert said yesterday.
Syria was also an important topic of discussion when US President Donald Trump called his Russian counterpart Vladimir Putin early this week.
According to a senior administration official, there is not a workable ceasefire in Syria now.
"We have a murderous Daesh, which has established control over areas through intimidation, coercion, the most brutal tactics that you can imagine, and crimes against humanity. You have a murderous regime that has committed mass murder of its own population, including with the use of chemical weapons - a regime that's backed by sponsors who have enabled that regime," the official said.
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"What we need is all parties to the conflict to recognise that the time is now to take a fundamentally different approach to this problem and begin to resolve the conflict in a way that stops the suffering of the Syrian people," the senior administration official said.
Russia is one of Assad's most important international backers and the survival of the regime is critical to maintaining Russian interests in the country.
The US has accused Assad of responsibility for widespread atrocities and says he must go. But it agrees on the need for a negotiated settlement to end the war and the formation of a transitional administration.
Since September 2014, the US has been conducting air strikes on ISIS and other jihadist groups in Syria as part of an international coalition against the dreaded jihadist group.