"Today we met with Mutko and we offered each other our mutual support," Lebedev said in an interview with the TASS news agency.
"He will support me in the RFU executive committee elections and I will support him in the presidential elections. Therefore, I will no longer be an opponent for him in the RFU elections," he added.
On Monday, the RFU registered both candidates to take part in the election, which is scheduled to take place on Sept. 2.
Mutko was previously in charge of the RFU from 2005-09, while Lebedev has never worked in a sporting capacity. In 2012, he sought election for the RFU presidency but was only able to muster nine votes.
The RFU has been looking for a new president since May 31, when Nikolai Tolstykh was removed from his position following a vote of no confidence. He was blamed for plunging the organization into substantial debt under his stewardship.
Russia has been without a coach since dumping Fabio Capello earlier this month and Mutko said a successor would be announced by Monday.
"Slutski and Borodyuk are the two coaches that will be chosen from," Mutko told the TASS news agency.
"Let's not rush into things and we will make the decision by Monday. I would prefer that the coach did not have any duties with a club side and I would want this coach to help us qualify for the European Championships," he said.
Slutski coaches CSKA Moscow while Borodyuk was deputy to Guus Hiddink and Dick Advocaat when they managed the Russian national team.