Hayward was called back to London a month ago after a bruising encounter with a Congressional committee and has since kept a low profile.
"We're getting to the end of the situation," said David Battersby at Redmayne Bentley Stockbrokers. "To draw a line under it, they need a new chief executive."
In New York, BP shares rose almost 5 percent Monday as the stock market anticipated a formal announcement about Hayward. Shares of BP PLC rose $1.82, or 4.9 percent, to $38.68 in midday trading in New York. BP shares closed up 4.6 percent Monday at 416.95 pence in London.
The BP board would have to approve a change in company leadership, and there is persistent speculation that chairman Karl-Henric Svanberg, who moved into the post on Jan. 1, is also likely to lose his job later this year.
The one-day board meeting comes a day before BP announces earnings for the second quarter. That report is expected to include preliminary provisions for the cost of the Gulf disaster, with analysts saying that could be as high as $30 billion.
"BP notes the press speculation over the weekend regarding potential changes to management and the charge for the costs of the Gulf of Mexico oil spill. BP confirms that no final decision has been made on these matters," the company said in a statement Monday to the London Stock Exchange before trading began.
Shares were up 2.6 percent at 408.95 pence ($6.33) in midafternoon trading in London.
Hayward, 53, who has a Ph.D in geology, had been a well-regarded chief executive. But his promise when he took the job in 2007 to focus on safety "like a laser" came back to haunt him after an April 20 explosion on the Deepwater Horizon rig killed 11 workers and unleashed a deep-sea gusher of oil.
Hayward's early attempts to shift blame to the rig operator, Transocean, failed to take the heat off BP. Later remarks that the amount of oil pouring into the Gulf was "tiny" compared to its volume of water and Hayward's whining that he would "like my life back" made him an object of scorn. That emotion turned to fury when Gulf residents heard that Hayward spent a day at a fancy English sailing race in which his yacht was competing at the height of the disaster.
David Cumming, head of U.K. equities at Standard Life Investments, said the board's reported intention to remove Hayward is an act of "political appeasement."
"I think they have taken view that his departure will relieve some of the political and media pressure in the U.S. and help BP rebuild its U.S. reputation," Cumming told BBC radio.
Chief executives inevitably often are sacked for corporate failure, whether or not they had any direct responsibility for what happened, said Howard Wheeldon, senior strategist at BGC Partners in London.
"Neither should we forget that Mr. Hayward has been master of his own downfall and that by those sometimes unfortunate remarks and attitude displayed in public he made his own situation all the more worse," Wheeldon said.
Dudley has so far avoided any gaffes. Currently BP's managing director, Dudley grew up partly in Hattiesburg, Mississippi. He spent 20 years at Amoco Corp., which merged with BP in 1998, and lost out to Hayward on the CEO's slot three years ago.
BP says the cost of dealing with the spill had reached nearly $4 billion by July 19, but that it was too early to quantify the eventual total cost.
New Orleans Mayor Mitch Landrieu said BP's attitude about making things right was more important than who is running the company.
"BP, from I think everybody's perspective, made a very bad mistake," he said. "I think what the world expects from BP is an acknowledgment that something was done wrong. I think BP has a long way to go to gain the trust of the people."
Hayward makes 1.045 million pounds ($1.6 million) a year as the company's head, according to its annual report. In 2009, he received a performance bonus of more than 2 million pounds plus other remuneration, bringing his total pay package to over 4 million pounds.
BP is the process of selling assets to raise $10 billion toward a $20 billion fund that will finance the clean up of the mess in the Gulf. BP announced last week that it had sold properties in the United States, Canada and Egypt to Apache Corp. for $7 billion.
Under pressure from President Barack Obama, BP has also announced that it will pay no more dividends to shareholders this year. That move disappointed some 18 million Britons, many of them retirees, who hold stock in what used to be the country's largest company.