Kommersant said the Russian partners could get 7.6% in BP, whose market value is estimated at $215.5 billion. Its Russian venture is estimated at $32.8 billion.
Viktor Vekselberg, one of four Russian investors in the joint venture, said as quoted by Kommersant that BP had rejected the proposal, as well as the investors' other demands on management and strategy.
The Russian oligarchs have demanded equal board representation, a cut in TNK-BP's foreign staff and the replacement of CEO Robert Dudley, who they accuse of putting British oil major BP's interests ahead of those of TNK-BP, including in oil projects abroad.
AAR - the consortium of Russian investors, also including German Khan, Mikhail Fridman and Len Blavatnik - said on Wednesday it would sue BP, which holds the other 50% in TNK-BP, for attempts to seize control in the joint venture.
BP Chairman Peter Sutherland said last week the Russian partners had resorted to methods used by Russia's corporate raiders to wrest control of the company.
Russian media reports on Monday quoted Vekselberg and Fridman as saying the asset swap was not longer being discussed.
"No way! The sale [of the 50% stake] is out of the question today. When the conflict is resolved, we will see," Vekselberg said.
Fridman said: "Any price [for the stake] proposed today would not correspond to the company's value in the future. TNK-BP has enormous growth prospects given the planned cuts in tax burden for oil producers in Russia and growing oil prices...," he said.
Set up in 2003, TNK-BP is Russia's third largest producer. It accounts for a quarter of BP's overall output.
Fridman said the Russian shareholders' proposals were still on the table, but no new talks have been scheduled.
AAR plans legal action in the international arbitration court in Stockholm and in Russia.
