United Nations Arab League envoy Lakhdar Brahimi attended talks with Russian Foreign Minister Sergei Lavrov.
Mr Brahimi is trying to build on an agreement reached in Geneva in June by world powers.
There have been international calls for the creation of a transitional government but left Assad's role unclear.
The mediator, who met Assad and others on a five-day trip to Syria, is to meet together with senior US and Russian diplomats in the coming weeks.
Two meetings so far this month have produced no signs of a breakthrough.
In Damascus last Thursday, Mr Brahimi called for a transitional government to rule until elections in Syria.
Mr Brahimi said only substantial change would meet demands of ordinary Syrians, but did not specify who could be part of the transitional body.
Russia has vocally supported Mr Brahimi's efforts while refusing to join Western and Arab calls for Assad's exit.
Syrian Deputy Foreign Minister Faisal Makdad discussed Mr Brahimi's proposals with Mr Lavrov in Moscow last Thursday.
Mr Lavrov yesterday urged the government and opposition to start dialogue.
Russia said it had invited the leader of the National Coalition opposition group to meet Russian officials in Moscow or elsewhere for the first time.
Russia, together with China, has angered the West and some Arab states by vetoing three UN Security Council resolutions.
The resolutions were meant to put pressure on Assad, who has given Moscow one of its firmest post-Soviet footholds in the Middle East.
President Vladimir Putin has said Russia is not trying to prop up Assad.
He contended that its vetoes and opposition to UN sanctions are driven by the principle of non-interference in sovereign states.
Russia has warned it will not allow a repeat in Syria of last year's events in Libya.
NATO intervention, authorised by the UN Security Council after Russia abstained from a vote, helped rebels topple Muammar Gaddafi.
Elsewhere, Syrian government forces have pushed rebels from a district in Homs after several days of fierce fighting in the strategically important city.
The army moved into Deir Ba'alba, a neighbourhood on the northeastern edge of Homs, they said, leaving the rebels controlling just the central neighbourhoods around the old city and the district of Khalidiyah, immediately to the north.
Homs, in central Syria, was the scene earlier this year of some of the heaviest fighting in the 21-month uprising against President Assad which has killed at least 45,000 people, according to activist tallies.