BP shares climbed 5% to 488.85 pence at 10:00 GMT, Reuters reported.
British newspaper The Daily Mail reported that Shell considered a bid, but was discouraged by the uncertainty over the size of the compensation claims that BP would face.
Shell has not ruled out making a bid for BP, The Daily Mail said.
"BP remains cheap and vulnerable at these levels but I do not think a bid is likely," the dealer said.
BP shares may also be benefiting from comments by Ken Feinberg, the official coordinating the compensation process for victims of the Gulf oil spill.
Feinberg said in a recent interview that BP will likely have to pay out no more than about half of the $20 billion it has set aside to cover compensation claims.
BP's shares have jumped 65% from a low of 296 pence. The company's' market value is now around $140 billion.