Such a deal would hand control of two major national UK newspapers to Lebedev, in addition to his acquisition of a 75.1% stake in the Standard in January.
According to sources, senior executives from Independent News & Media (IN&M) have been in discussions with Lebedev's representatives over a possible sale, among several options. According to one source close to the situation, a nominal offer for the two titles has already been tabled and a deal could be finalised by the end of the month.
The two parties have been in discussions over a number of possible deals since the Independent moved into the same Derry Street offices as the London Evening Standard in May, according to sources.
It is thought negotiations with IN&M are so advanced that discussions have already been held over the new management structure for a combined entity.
One source said: "There have been a number of discussions between the two parties, one of which has been over the sale of the Independent and Independent on Sunday to Lebedev."
Denis O'Brien, a 26% shareholder in IN&M, has previously called for executives at the media group to offload the titles. Pre-tax profit at IN&M's UK operations slumped nearly 99% year on year in 2008 to just €200,000, following a drop in circulation and ad revenue.
But sources claim IN&M will not sell the titles until the group has resolved the refinancing of its €220m bond, which expires on 26 June.
A representative for Lebedev Holdings Ltd was unavailable for comment. IN&M and the London Evening Standard declined to comment. The Standard's former owner, Daily Mail & General Trust, which retains 24.9% ownership of the paper, was also unavailable for comment.