The move would trigger a further shareholding reshuffle at the consortium after another member, Gulf Arab state Oman, said it was also looking to sell its stake.
Most of the shareholders of the Chevron-led pipeline, which runs to the Black Sea port of Novorossiisk, have agreed on the expansion terms demanded by Russia, which owns 24 percent in the consortium as a host state.
BP, the only shareholder that still opposes the terms, said it was considering selling the stake if no compromise was found.
"This is one of the options to settle the current situation," said Vladimir Buyanov, a BP spokesman in Moscow.
He said BP could sell its stakes in LukArco and Kazakhstan Pipeline Ventures, which are members of the consortium. BP's stakes in the ventures bring its share in CPC to 6.6 percent, Buyanov said.
Pipeline monopoly Transneft, which holds the country's stake in CPC, had long opposed the plan to double the pipeline's capacity from the current 700,000 barrels per day, but it has now dropped its objections.
Transneft previously argued that the pipeline yielded low returns and that expansion would add pressure on the already congested Turkish Straits shipping route.
In summer, most of the partners agreed to raise the shipping tariff to $38 per ton from $30.24 last year, and private investors agreed to halve interest rates on a $5 billion loan to CPC to 6 percent, easing worries over funding.
Transneft, which owns all pipelines on the Russian territory except CPC, has said BP was insisting on borrowing more to fund the expansion.
A London-based source close to BP told Reuters on Thursday that BP wanted to borrow more, as its percentage interest in the pipeline was bigger than its percentage interest in the Kazakh fields, which feed the route.
This means BP has more incentive for the pipeline project to be commercially attractive, the source said.
Russia and Kazakhstan, which is also a state shareholder of CPC, have both expressed interest in buying Oman's 7 percent stake.
Besides BP and Chevron, which holds 15 percent, its private shareholders include Shell, ExxonMobil and Russia's two largest oil producers, Rosneft and LUKoil.
CPC has been shipping oil since 2001 and pumps up to 750,000 barrels per day to Novorossiisk for re-export to the Mediterranean.
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