Ex-Navy Seal Robert O’Neill, 38, has told the Washington Post in a new interview he fired the fatal shot.
This contradicts the account of Matt Bissonnette, another former Seal involved in the raid, in a 2012 book.
The al-Qaeda leader was killed in a 2011 Navy Seal raid on his compound in Abbottabad, Pakistan.
Mr O’Neill, who retired in 2012, had previously told his story anonymously to Esquire magazine.
He was scheduled to reveal his identity in a television interview later this month, but a website published Mr O’Neill’s name pre-emptively in protest of his decision to go public, the Post reports.
Mr O’Neill said he and another member of the team whose identity remains secret climbed to the third floor of the compound in Abbottabad, Pakistan and saw Bin Laden poke his head outside the door to one of the rooms.
The unnamed commando, at the point position of the formation, fired at him but missed, according Mr O’Neill.
An instant later, Mr O’Neill went into the room and killed the al-Qaeda leader with shots to the head, he says.
However, in the book No Easy Day, Mr Bissonnette claimed it was the point man who killed Bin Laden.
On Thursday, Mr Bissonnette did not directly dispute Mr O’Neill’s claim, in an interview with NBC News.
“Two different people telling two different stories for two different reasons,” Mr Bissonnette told the broadcaster.
“Whatever he says, he says. I don’t want to touch that.”
Mr Bissonnette is scheduled to appear on the CBS news magazine programme 60 Minutes ahead of the publication of his second book, No Hero, about his service with the Seals.
Meanwhile, he is under investigation for potentially disclosing classified information in his first book, about the Bin Laden raid.
The official account of what happened is unlikely to be disclosed by the US government for many years.
Pentagon officials have neither confirmed nor denied Mr O’Neill’s account, but senior special operations leaders sent a letter last week to all Navy Seals urging them to comply with their code of silence about operational details, including avoiding taking “public credit”.
“We do not abide wilful or selfish disregard for our core values in return for public notoriety and financial gain,” they wrote.
Bin Laden was confirmed killed in the raid and his body was buried at sea.
Darkness and close quarters inside the compound have made some Navy Seals question whether it is possible to determine whose bullets killed the al-Qaeda leader.
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