BPconfirmed to CNBC thatit's in substantive talks with potential buyers about selling some of its assets.
The company is not in discussions, however, to be acquired by Exxon, BP spokesman Andrew Gowers said.
BP also said Monday that the cost of dealing with the oil spill in the Gulf of Mexico has risen to $3.5 billion, though its shares rallied on earlier reports it may sell some assets.
The oil company said the overall cost includes nearly $165 million paid to settle individual claims. BP said that by Saturday, it had received 105,000 claims and made more than 52,000 payments.
The company says it is still too early to estimate the final total of costs and compensation.
BP shares, meanwhile, continued their recent rally, climbing in early trading on the London Stock Exchange .
BP's London stock, which was trading at 655 pence before the Deepwater Horizon drilling platform disaster on April 20, had briefly fallen below 300 pence last month.
In New York, BP shares closed higher Friday.
Over the weekend, The Sunday Times in London reported that BP was talking with Apache about selling 12 billion pounds ($18 billion) in assets including a stake in Alaska's Prudhoe Bay field.
The Sunday Times also reported that ExxonMobil had approached U.S. government officials asking if they would object to a takeover bid for BP. None of the companies would comment on the reports.
"We continue to see good near-term upside in the shares, most particularly if the first relief well is successful in capping the Macondo well," Collins Stewart analyst Gordon Gray said Monday in a research note.
BP reports its second quarter results on July 27, a date which some commentators have suggested as a deadline for announcing significant steps toward dealing with the costs of the Gulf of Mexico disaster.
Gray believes BP may come through the second quarter with no significant change in net debt, particularly as the company has canceled dividend payments for the rest of the year in response to pressure from U.S. President Barack Obama.