Lebedev bought the Evening Standard title from Associated Newspapers last year and was widely believed to be interested in adding The Independent to his portfolio.
At that time, rumours came to nothing, but the Guardian has reported today that the parties are in negotiations over a possible deal.
According to a source quoted in the Guardian, "there is confidence a deal can be done soonish".
Reports claimed that the possible deal arose after major shareholders Denis O'Brien and Gavin O'Reilly put a long-running squabble behind them.
The Evening Standard and The Independent titles are both based at DMGT's Derry Street headquarters in Kensington.