BP said on Monday that "no final decision" had been made about Hayward.
Public relations gaffes
BP capped the leaking well last week, choking off the flow of oil for the first time since an April 20 explosion killed 11 workers on a drilling rig and sent crude gushing into the Gulf.
Hayward has been criticised for a string of public relations gaffes during the crisis included telling reporters "I want my life back".
Monday's meeting comes a day before the release of second-quarter results, which are expected to reveal a $30bn provision for funding the oil disaster.
Dave Kansas, the Wall Street Journal's European markets editor in London, told Al Jazeera that Bob Dudley, BP's managing director, was tipped to take over the job of chief executive.
Kansas said: "He [Dudley] grew up on the Gulf coast in Mississippi ... he's received high praise from some of the government officials around the BP situation already even though he hasn't taken the job.
"The PR piece of this looks like it's probably sliding into a place that BP would like.
He added that having an American take the chief executive slot was also important for the company.
'Refining the operation'
The news of Hayward's possible departure came as US officials said that BP would finish installing the last bit of pipe into a relief well engineered to help permanently plug its oil spill in the Gulf of Mexico in the coming week.
Once the pipe is cemented in place, BP will begin the "static kill" sealing process in the first week of August, Thad Allen, who is heading the US government's response to the spill, said on Sunday.
Allen had previously said that the "static kill" technique could start in three to five days after the pipe was fully installed.
But he said on Sunday that the timeline for the operation had been "refined" to be more conservative after consulting with BP.
He also said Tropical Storm Bonnie, which had prompted many ships to evacuate the spill site and delayed work on the relief well off Louisiana's coast, could push back BP's mid-August target date for completing the plugging process by seven to nine days although the vessels returned to the site.
The operation aims to seal the well by pumping heavy drilling mud through the blowout preventer valve system that sits on top of the well and then injecting cement inside to seal it.