BP PLC’s board is set to name managing director Bob Dudley as the company’s new chief executive if, as expected, the board on Monday approves the negotiated departure of current chief Tony Hayward, people familiar with the matter said.

The plan before the board would elevate Mr. Dudley to the CEO spot on Oct. 1, allowing for a roughly two-month transition from Mr. Hayward, according to someone familiar with the plan. Mr. Hayward would stay on the board for the rest of the year. As expected, he will discuss BP’s second-quarter results on Tuesday.

The decision to move toward Mr. Hayward’s departure was “mutual,” the people familiar with the matter said. Mr. Hayward has been under heavy fire for his handling of the massive Gulf of Mexico oil spill that followed an April explosion and fire on the Deepwater Horizon oil rig.

Since then, the company has faced a torrent of criticism for both its safety practices and its response to the spill. Since taking over as CEO, Mr. Hayward had repeatedly claimed that he had taken strong steps to improve practices that had led to past safety incidents.

In the U.S., Mr. Hayward became a lightning rod for criticism after his poor performance before a congressional panel and an infamous gaffe in which he complained, “I want my life back.”

Mr. Dudley is a BP board member who, as managing director, has oversight of its activities in the Americas and Asia. He spent nearly two decades working for Amoco before it was acquired by BP in 1998, and in 2007 nearly became BP’s chief executive, losing a battle with Mr. Hayward to succeed CEO John Browne.

By turning to Mr. Dudley, BP would be installing an American at the helm of the British company as it battles to restore its reputation and credibility.

BP and Mr. Hayward have been criticized for not being attuned enough to the way the spill has impacted the Gulf region. Mr. Dudley, on the other hand, grew up in the region, in Hattiesburg, Miss.

Earlier this summer, with Mr. Hayward taking one body blow after another in the first weeks of the crisis, he was sidelined and Mr. Dudley become the public face of the company in the region.

The elevation of Mr. Dudley may be something of a surprise, even to him. Inside BP, he wasn’t seeking the CEO post, says one person familiar with the situation

In the middle of last week, Mr. Dudley flew to the corporate headquarters in London—and stayed rather than quickly returning to the Gulf Coast as he planned, according to one BP manager. Mr. Dudley’s focus when he arrived was to help quantify the massive liability the company would need to report in its second-quarter earnings Tuesday. Yet, in recent days, the topic shifted to his becoming the next CEO.

Before this latest development, Mr. Dudley has worked nearly around the clock to gear up the new Gulf Coast restoration unit to which he had been appointed. In the last two weeks, he met individually with the Gulf Coast governors, and flew by helicopter to the site of the runaway well with the administration’s point person on the spill, Adm. Thad Allen.

He also worked to select executives for the Washington and Houston offices, and picked a new chief financial official for the unit he appeared intent on beefing up.

Mr. Hayward’s departure, if the board agrees, would show the oil giant is taking strong action to deal with one of the largest oil spills in U.S. history from its Gulf of Mexico runaway well. With a new cap appearing to hold the oil for now, the talks between Mr. Hayward and BP intensified in recent days.

Mr. Hayward has spent the last week or so weighing whether he should resign “with honor for the sake of BP,” says one person familiar with his thinking.

The cost of the massive oil spill to BP has been enormous—and is continuing. An internal tally shows work at the sub-sea level to contain the oil and the Gulf Coast shoreline to clean up the oil is costing about $100 million a day.

That doesn’t include the expected flurry of claims, for which the Obama administration pressured BP to set up a $20 billion escrow fund. The company last week agreed to sell $7 billion in assets.

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