The latest estimate is far higher than the $2.65bn that was given one week ago.
Britain's Sunday Times newspaper said BP's advisers were trying to drum up interest among rival oil groups and sovereign wealth funds to take a stake of between five and 10 percent in the company at a cost of up to $9.1bn.
Such an investor would help ward off a takeover and raise funds for the liabilities generated by the worst oil spill in US history, the reports said.
Shares in BP closed up 2.16 per cent in London on Monday, off earlier intraday peaks on the reports.
BP declined to comment on the reports and some shareholders balked at the idea of a strategic investor.
"We don't think a strategic partner is at all necessary," said one top-10 BP shareholder who did not want to be named.
"We think this is just people trying to panic the company and stampede into doing something to earn huge fees from selling new shares in BP.
"Shareholders will be saying 'No, thank you' to this and we have communicated this to the company."
Another top-10 investor, who also wished to remain anonymous, agreed that BP "probably did not" need a strategic investor at the moment.
One former investment director at a Dubai-based state investment company said it was a predictable move for BP but he doubted the company would find a taker among the oil rich sovereign wealth funds (SWFs) of the Middle East.
A London banker said any SWF involvement might more likely involve the Far East than the Middle East.
|Clean-up workers have arrived back on Grand Isle, Louisiana, by the hundreds [AFP]
Separately, several newspapers reported interest among SWFs in buying some of BP's assets in the Middle East and Asia.
BP has said it hopes to raise $10bn from asset sales this year as part of its plan to fund a $20bn clean-up fund set up under pressure from US authorities.
Al-Jarida, an Arabic language daily newspaper, was most specific, saying Kuwait Foreign Petroleum Exploration, a state-run firm, is reviewing investing in BP oil fields in Egypt, Yemen and East Asia.
Al Jazeera's John Terret, reporting from New Orleans, said that the speculation was caused by doubts over whether BP can keep losing money at such a rate.
"BP says it has released more than 30,000 cheques in the last month alone," he said.
"The question is whether the company can survive at that rate of expediture."
Meanwhile, efforts to clean up the spill in the area, where a ruptured well has been spewing crude since April 20, gathered steam on Monday after Hurricane Alex prompted a five-day shutdown.
Clean-up workers arrived back on Grand Isle, Louisiana, by the hundreds, spilling off school buses that shuttled them in from around the state.
In the wake of the hurricane, beaches, shorelines and marshes lay smeared with thick patches of oil and the sky was still filled with ominous grey clouds.
Tests on a supertanker adapted to skim large quantities of oily water from the surface were inconclusive because of high seas, ship owner TMT Shipping Offshore said.
The M/V "A Whale" is seen as a potential saviour of efforts to clean the oil pollution because it can collect 500,000 barrels per day of contaminated water.
Tests on the so-called super skimmer conducted just north of the blown-out BP well were supposed to be completed on Monday but have been extended because of the weather, Bob Grantham, TMT's spokesman, said.
"After an initial 48-hour testing period results remain inconclusive in light of the rough sea state we are encountering," he said.