The Attorney General is being asked to consider bringing perjury charges against the former boss of BP.
Lord Browne dramatically quit as the oil giant's chief executive on Tuesday after he admitted lying to a court during a battle to block stories about his private life.
He could lose up to £15.5m after failing in a legal battle to keep details of his four-year relationship with Canadian Jeff Chevalier private.
Lord Browne said it was "a matter of deep regret" that he lied over how he had first met Mr Chevalier in the course of his fight with Associated Newspapers, which publishes the Mail on Sunday.
He added that he was resigning early - he had been due to step down in the summer after 41 years at the firm - to "avoid unnecessary embarrassment and distraction" to BP.
The ex-BP boss, who has lost a £3.5m leaving package and a potential £12m in share options, said in a statement: "I have always regarded my sexuality as a personal matter, to be kept private.
"It is a matter of personal disappointment that a newspaper group has now decided that allegations about my personal life should be made public."
But the Mail on Sunday said it was Lord Browne who had made his private life a public issue by lying, not the newspaper.
The Mail on Sunday also pledged to make its evidence on Lord Browne's deception available to Attorney General Lord Goldsmith.
Lord Browne's designated successor, Tony Hayward, immediately took over as BP's chief executive yesterday. BP's shares closed 2.5p down at 563p.
