The UK company put its 50pc stake in TNK-BP, its main Russian venture, up for   sale on June 1 after a breakdown in relations with oligarch partners AAR.

Igor Shuvalov, Russian first deputy prime minister, said he would be happy to   see AAR buy out BP, or for AAR to find a new partner.

But he added: “In the big picture, the only thing I would regret is that BP would leave. Having such an investor on the market is a very valuable thing.”

Mr Shuvalov told the Wall Street Journal he would not want to see a   state company such as Rosneft buy BP’s stake “except as part of a deal to   bring in another foreign partner or as part of a larger transaction to   privatise Rosneft”.

BP and Rosneft attempted a share-swap partnership deal last year that would   have seen BP take a 9.5pc stake in Rosneft.

Mr Shuvalov appeared to hint he could be amenable to a similar deal now,   saying it would be “ideal” if “major global energy companies were   shareholders of Rosneft”.

Meanwhile, it emerged that AAR is mulling a bid for half of BP’s 50pc stake in   TNK-BP, to give it control of the venture.

Sources with knowledge of AAR’s position said an offer for the 25pc stake   could be in the region of $7bn (£4.4bn), taking into account potential   multi-billion dollar damages AAR is thought to be seeking from BP over the   attempted Rosneft deal.

It would also imply a valuation of BP’s stake of about $20bn, well below   consensus estimates of $25bn to $30bn. Analysts at Citi said that, after   meeting with AAR , they believed a $25bn sale was “unrealistic” and that   “AAR is likely the only realistic bidder for BP’s stake”.

A BP spokesman said “AAR are not the only potential purchaser”, adding “no   value whatsoever” should be attributed to “any supposed damages claim”.

Mikhail Fridman, the Russian billionaire, has been meeting institutional   investors in London to canvas support for his plan to buy half of BP’s stake   in TNK-BP, its Russian joint venture, according to the Financial   Times, citing people familiar with the matter.

A report in The Times claims BP may be   forced to sell a stake in the joint venture to partners at a knockdown price   after AAR threatened to block a sale to any other company.