The group includes opposition parties of various ideologies, including Arab and Kurdish nationalists, Marxists and independent figures such as writer Michel Kilo and economist Aref Dalila.
Syria has been rocked by protests against Assad's regime that began on March 15 and triggered a brutal crackdown in which the United Nations says 2,600 people have been killed.
A delegation of Russian lawmakers is in the country in a bid to broker an end to the violence.
Assad told them he welcomes the "balanced and constructive Russian position toward the security and stability of Syria," the state-run SANA news agency reported.
But Assad, who has blamed "armed terrorist gangs" for the violence rocking his country, also warned against "any foreign intervention that threatens to divide states in the region."
Ilyas Umakhanov, deputy head of Russia's upper house of parliament, said "the country's leadership understands that one can only overcome a political crisis by uniting all the country's healthy political forces," Russia's Interfax news agency said.
"We once again saw for ourselves that the country's leadership intends to firmly move along the path of political reforms, create all the necessary conditions to consolidate society and all the patriotic forces of the country," Umakhanov was quoted as saying after meeting Assad.
Russia has opposed efforts led by the United States and the European Union to push for a UN Security Council resolution condemning the Syrian regime over its deadly crackdown on pro-democracy protesters.
The opposition, meanwhile, is trying to unite against the regime.
Opponents plan to announce the formation of a coalition that includes the Coordinating Committee, liberal parties of the opposition "Damascus Declaration," the outlawed Muslim Brotherhood and independent Islamists.
Two opposition groups were SET up in Turkey at the end of August: the mostly Islamist "National Council" and the "National Council of Syrian Transition" headed by Burhan Ghaliyoun, a Paris-based academic.
"For the overthrow of the tyrannical and corrupt security regime and for democratic change, the peaceful revolution of the Syrian people must continue," said a statement read Sunday by Abdel Aziz Khayer of the Coordinating Committee.
"We must end the military solution, allow peaceful protests, withdraw the army to the barracks, try those responsible for the massacre of protesters, release all political prisoners and begin reconciliation between the army and the people," it added.
Another committee member, Rajaa Nasser, said that "all movements of the Syrian opposition agree on the need for change. The majority reject any military intervention" in Syria, he added.
Samir Aita, editor of Le Monde Diplomatique in Arabic and European representative of the Coordinating Committee, announced a September 23 meeting in Berlin.
"It is necessary to unify (opposition) efforts for the change to happen," he said, adding that it was important that the various opposition groupings should "unite around common goals."
Saturday's meeting elected an 80-member central council, 25 percent of them "young revolutionaries."
Meanwhile, the Syrian Observatory for Human Rights reported that four people, including an 11-year-old boy, shot in recent raids died on Sunday of their wounds.
As schools reopened, students demonstrated against the regime in the central city of Homs and security forces arrested 70 civilians, relatives of people wanted by the regime, in the northwestern province of Idlib, it said.
The official SANA news agency also reported that the annual Damascus film festival SET for October has been cancelled because of the unrest.