LONDON — In a last-ditch effort to try to get Parliament to pass her plan for Britain to leave the European Union, Prime Minister Theresa May on Wednesday offered to step down and allow another prime minister who has the confidence of her party and lawmakers to negotiate the final details.

Mrs. May’s stunning overture to her fellow Conservatives came just as Parliament tried to sideline her and come up with its own plan for Brexit, as the process of leaving the bloc is known.

But when lawmakers held a series of nonbinding votes on Wednesday night on eight different options for Britain’s future relationship with the European Union, none mustered a majority.

Mrs. May is so unpopular and has lost so much authority within her party that her offer to step down, if her plan is approved, was greeted with relief by Tory lawmakers.