The Kremlin declined to comment immediately on the report, as did Saudi Arabia's embassy in Moscow.
Saudi Prince Bandar bin Sultan met Russian President Dmitry Medvedev and Prime Minister Vladimir Putin in Moscow for talks on Monday that focused on widening bilateral cooperation.
Kommersant said Saudi Foreign Minister Prince Saud al-Faisal suggested Russia curtail its cooperation with Iran at a meeting in Moscow this February. The proposal was made to the Kremlin in the name of King Abdullah.
"The Kingdom's leadership advised Moscow to phase out cooperation with Tehran and in return promised attractive contracts with Saudi Arabia," said the paper, one of Russia's most respected dailies.
"In essence, Russia was offered to become a major partner in the Middle East."
The paper said Bandar, who heads Saudi Arabia's National Security Council and is an influential former ambassador to Washington, fleshed out the proposal when he met Kremlin leaders this week.
Saudi Arabia, the world's top oil producer and a close U.S. ally, is wary of Iran's ambitions and shares Western concerns that Tehran is seeking to develop nuclear weapons. Iran says its nuclear programme is peaceful.
Russia, which has tried to boost cooperation with Tehran, says there is no evidence Iran wants to build a nuclear weapon but Moscow fears a conflict in the region would stoke instability close to its southern borders.
Russia, awash with oil cash, has been trying to increase its influence in the Middle East after the chaos which accompanied the fall of the Soviet Union.
Moscow is helping Tehran build its first nuclear power station and Putin, while still president, visited both Riyadh and Tehran last year.
It was the first visit by a Russian leader to the Saudi kingdom and marked closer ties since a 2003 visit by King Abdullah, then crown prince, to Moscow.
Saudi Arabia is interested in buying air defence systems, helicopters and tanks from Russia, Kommersant said.
"Saudi Arabia strives to have varied sources of arms," Bandar was quoted as saying by Russian news agencies on Monday after talks with Putin.
(Additional reporting by Oleg Shchedrov, editing by Dominic Evans)
