He is due to meet Vladimir Putin, the Russian president, on Tuesday for their first talks since Hamas seized control of Gaza on June 14.
Moscow's official contacts with Hamas have riled Abbas, who will likely press Putin to sever links, analysts said.
Abbas said he did "not object to Russia ... mediating between [Abbas's movement] Fatah and Hamas".
In an interview that was aired by Russian state television on Monday, he said: "The most important thing that I will raise and discuss with President Putin ... is the bloody overthrow that happened in Gaza."
Abbas said he would also discuss conditions for holding an international conference on the Middle East.
He said he would inform the Russians about talks with Tony Blair, the former British prime minister and the quartet's new envoy.
Russia, along with the US, EU and the UN, make up the Quartet.
Russia is the only member that has ties with Hamas, which is considered to be a terrorist group by the US and EU.
'Mass of questions'
"There are a mass of questions we want to discuss with Russia's leadership including who will participate in the conference and what questions will be discussed there," Abbas said.
Last week, Lavrov had a telephone conversation with Khaled Meshaal, Hamas's political chief.
Since Hamas won a Palestinian parliamentary election last year, the US and the EU have sought to bolster Abbas and sideline Hamas financially and diplomatically.