The latest round of talks began days after the Islamic State jihadist group was forced out of its de facto capital Raqa in northern Syria, in a major victory for the US-backed Kurdish-Arab Syrian Democratic Forces.
Russia's military intervention in 2015 turned the tables in Syrian president and Kremlin ally Bashar al-Assad's favour, but the role of diplomatic arbiter in the complicated conflict is a new one for Moscow.
Assad "has confirmed his readiness for... the preparation of a new constitution and the holding of new parliamentary and presidential elections on this basis", Lavrentyev said today.
Lavrentyev called Assad's acceptance earlier this month of a constitutional reform process "a very important announcement".
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He also said Russia is ready to host a "congress of the peoples of Syria" involving both regime and opposition representatives.
The precise date and location of the congress would be determined in Astana, Lavrentyev said, suggesting it could be at Russia's Hmeimim military base in Syria.
Recent rounds of talks in the Central Asian nation have focused on ironing out the details of a Russia-led plan establishing four de-escalation zones in Syria.
The plan was first tabled in Astana in May to minimise fighting between government forces and moderate rebel factions and improve civilians' access to aid.
The most recent round of talks in September saw Russia, Turkey and Iran agree to jointly police a buffer zone in the contentious northern province of Idlib, where Ankara and Tehran are viewed as having competing interests.
Despite backing opposite sides in the war, Ankara and Moscow have been working closely on Syria since 2016, following a rift caused by Turkey shooting down a Russian war plane.
The Russian foreign ministry confirmed that the country's top diplomat Sergei Lavrov discussed the Syria talks today in a telephone conversation with his Turkish counterpart Mevlut Cavusoglu.
The Russian ministry said the diplomats confirmed their "mutual intention to continue efforts, including at international meetings, in the interests of further progress in the talks process between Syrians".
Russian military police are present in all four zones.
Syria's United Nations envoy Staffan de Mistura urged parties to move towards "a more stable political settlement" during a mid-October visit to Moscow.
While the de-escalation zones brought about an initial reduction in fighting, the International Red Cross has voiced concern that the violence has intensified again, including in the zones.
Russian President Vladimir Putin will travel to Tehran on Wednesday and will hold bilateral talks with Iranian President Hassan Rouhani, the Kremlin said today.