Earlier Saturday French President Francois Hollande signed into law a controversial bill allowing same-sex marriage. The bill has sparked mass protests throughout the nation.
Russian President Vladimir Putin said late last month Russia could change its adoption agreements with the countries which allowed gay marriage that goes against "traditional Russian values".
The Russian law defines marriage as between one man and one woman.
"France has other variants now. It turns out, that the countries (Russia and France) have a different legal framework. It is obvious, that a new mechanism should be devised to resolve this issue," Astakhov told the Echo Moskvy radio station.
Russia's State Duma, the lower house of parliament, has been requested to come up with amendments to the law by July 1, Astakhov said.
Homosexuality was only decriminalised in Russia in 1993, and anti-gay sentiment remains strong in society, including among Russia's political establishment.