One of America’s most powerful families will call tomorrow for a sweeping shake-up at the top of ExxonMobil, the world’s largest company.

A group of descendants of John D. Rockefeller, who founded Exxon's predecessor Standard Oil in 1870, will begin a campaign to split the role of chief executive and chairman of the board at the oil and gas group, a role held by Rex Tillerson.

Last night the family group issued a statement saying that the company’s leadership was “failing to address the future of energy and related industry hurdles.”

It said that representatives would make an announcement in New York to explain “that a majority of the family is now so concerned about the direction of ExxonMobil Corporation that it is urging a major change.”

Exxon, which earned $40 billion (£20 billion) last year, when Mr Tillerson was paid $21.7 million, was the slowest of the big oil majors to acknowledge climate change. The family is calling for an independent chairman and a bigger leadership role for the directors. The campaign comes as big oil companies face mounting pressure to deal with public concern over global warming.

More than 100 Rockefeller descendants hold a significant stake in Exxon through a variety of trusts, but the exact percentage is unknown.

The campaign is being spearheaded by Neva Rockefeller Goodwin, an economist and great-granddaughter of John D. Rockefeller, as well as Peter O’Neill, head of the Rockefeller family committee dealing with ExxonMobil. He is a great-great-grandson of John D. Rockefeller.

Exxon produces nearly 4.2 million barrels of oil a day and had revenues of $404.5 billion during the past fiscal year. Its market capitalisation is about $500 billion.

In 2006 Senator Jay Rockefeller wrote to Mr Tillerson urging the company to stop funding groups that denied the existence of climate change.

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