The relevant resolutions were announced during rallies in the two separatist republics' capitals earlier Thursday.
In Sukhumi, around 50,000 people gathered on the central square to show support for the appeal to recognize Abkhazia "as a sovereign and independent state, and sign a treaty of friendship, cooperation and mutual assistance."
"It has become clear that Abkhazia will not live in the same state with Georgia," Abkhazian parliament speaker Nugzar Ashuba told the rally.
Addressing a similar gathering in Tskhinvali, South Ossetian President Eduard Kokoity said: "Those who armed Georgia are also responsible for what happened in South Ossetia, and do not have the moral right to claim the role of peacekeepers."
"We appeal to Russia to be the first country from the international community to recognize the independence of the republic of South Ossetia," reads Tskhinvali's request.
Both chambers of Russia's parliament are expected to consider the appeals by the republics on Monday.
Russian Foreign Minister Sergei Lavrov signaled that whether Russia recognized Abkhazia and South Ossetia as independent countries would depend on the Georgian president.
"[Mikheil] Saakashvili is responsible for how the situation will develop," Lavrov said
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The main event of the third day of the 11th meeting of the Valdai International Discussion Club in Sochi was the closing session with President Vladimir Putin. The atmosphere was calm and open, despite the current political tensions and the Russia-West confrontation. The Russian president said that it corresponded to the spirit of the Valdai Club.