"I do not think that Tony Blair will become the Quartet's conductor. His role will consist of following the U.S. line," said Yevgeny Primakov, who is also chairman of the Russian Chamber of Commerce and Industry.
He said that the Quartet's role has weakened considerably since its creation.
Primakov said that in the situation now prevailing, the Quartet should work out a compromise formula jointly with representatives of the conflicting sides - the Palestinians and Israelis - and also with the participation of other Arab countries, making it incumbent on the conflicting sides to fulfill this resolution.
"The Quartet should declare that if you reject this decision, all of its members, who represent great powers, will pull out of the peace process," he said.
Blair formally resigned as Britain's prime minister in late June after more than a decade in office, and within hours was confirmed as the new envoy of the Quartet of Mideast mediators, comprised of the United States, Russia, the European Union and the United Nations. The Arab League and the government of Israel both welcomed his appointment.
Blair's controversial Mideast track record reportedly prevented him from securing universal support in the Quartet. The British newspaper The Guardian and other Western media said earlier that Russia was trying to block his candidacy, a charge denied by Russian officials.
The Quartet said the duties of its new envoy would involve building government institutions for the Palestinians as well as promoting their economic development and raising aid.
Palestinian Prime Minister Salam Fayyad also welcomed Blair's appointment as Mideast envoy, and Palestinian negotiator Saeb Erekat said the former British premier would be an asset to Palestinian attempts to establish an independent state alongside Israel.
But representatives of the radical Islamist movement Hamas, which took over the Gaza Strip three weeks ago, expressed skepticism about Blair's ability to advance a two-state solution, describing him as an ally of Israel and the United States.
