The proposals were put forward by international envoy Lakhdar Brahimi, who has had talks with Mr Assad and is planning to hold meetings with Syrian officials and dissidents in Damascus.
He is trying to broker a peaceful transfer of power, but has disclosed little about how this might be done.
More than 44,000 Syrians have been killed in a revolt against four decades of Assad family rule, a conflict that began with peaceful protests but which has descended into civil war.
Past peace efforts have floundered, with world powers divided over what has become an increasingly sectarian struggle between mostly Sunni Muslim rebels and Mr Assad's security forces, drawn primarily from his Shi'ite-rooted Alawite minority.
Deputy Foreign Minister Faisal Makdad flew to Moscow to discuss details of the peace proposals, said a Syrian security source, who would not say if a deal was in the works.
However, a Lebanese official close to Damascus said Mr Makdad was to seek Russian advice on a possible agreement.
He added Syrian officials were upbeat after the talks with Mr Brahimi, the UN-Arab League envoy, who has not outlined his ideas in public.
The official said: "There is a new mood now and something good is happening."
Russia, which has given diplomatic and military aid to help Mr Assad weather the 21-month-old uprising, has said it is not protecting him, but has criticised any foreign backing for rebels and, with China, has blocked UN Security Council action on Syria.
A Russian Foreign Ministry source said Mr Makdad and an aide would meet Foreign Minister Sergei Lavrov and Mikhail Bogdanov, the Kremlin's special envoy for Middle East affairs, today, but did not disclose the nature of the talks.
Given the scale of the bloodshed and destruction, opposition groups insist Mr Assad must step down. Moaz Alkhatib, head of the Syrian National Coalition, has criticised any notion of a transitional government in which Mr Assad would stay on as a figurehead president stripped of real powers. Comments on his Facebook page have suggested the opposition believed this was one of Mr Brahimi's ideas.
Mr Alkhatib said the national coalition had told Mr Brahimi it rejected any such solution.
In other developments, Syrian army shelling killed about 20 people, at least eight of them children, in the northern province of Raqqa, say campaigners.
A video, published by the Syrian Observatory for Human Rights, showed rows of blood-stained bodies laid out on blankets. The sound of weeping could be heard in the background. The shelling hit the province's al-Qahtania village, but it was unclear when it had occurred.
Meanwhile, the head of Syria's military police has changed sides and declared allegiance to the anti-Assad revolt. In in a video on YouTube, General Abdelaziz Jassim al-Shalal said: "I have defected because of the deviation of the army from its primary duty of protecting the country and its transformation into gangs of killing and destruction."
A Syrian security source confirmed the defection, but said Mr Shalal was near retirement and had defected to "play hero".
Syrian Interior Minister Mohammed Ibrahim al Shaar is returning to Damascus after being treated in Lebanon for wounds sustained in a rebel bomb attack earlier this month.