LONDON — Prime Minister Theresa May asked the European Union on Friday to delay Britain’s departure from the bloc for a second time — until June 30 — and conceded that the country was preparing to take part in European Parliament elections.

Mrs. May made a formal request in a letter to Donald Tusk, the president of the European Council, for a postponement of the departure, now scheduled for April 12, but analysts said her proposed date was likely to be rejected by Brussels.

British news reports said that Mr. Tusk was pushing European leaders to offer Mrs. May a one-year extension while leaving the door open to an earlier withdrawal if Britain ratifies a deal for Brexit, as the process is known.

That plan, described in Brussels as a “flextension,” would eliminated the need for European leaders to repeatedly consider British requests for a delay. And in allowing Britain to leave sooner if an agreement is reached, Mr. Tusk appears to be trying to make it clear that Brussels is not trying to trap Britain in the bloc.

Mr. Tusk’s plan would need the backing of the leaders of European Union member states.

In asking for an extension until June 30 — the same date she once asked for but which the European Union previously rejected — Mrs. May was bowing to pressure from within her Conservative Party not to be seen as forcing the country into a longer delay. But she was also laying the ground for a more protracted extension by agreeing that Britain was prepared to participate in European elections in May. That was seen in Brussels as a condition for another Brexit postponement.

Mrs. May has sought over the past week to break months of deadlock by meeting with the leader of the opposition Labour Party, Jeremy Corbyn, to try to reach an agreement. But she said in her letter to Mr. Tusk that if those talks did not produce a compromise, she would hold a series of votes in Parliament on alternative paths in the hopes that lawmakers would eventually settle on one.

“This impasse cannot be allowed to continue,” Mrs. May wrote. “In the U.K. it is creating uncertainty and doing damage to faith in politics, while the European Union has a legitimate desire to move on to decisions about its own future.”

The prime minister’s Brexit deal has already been rejected three times by British lawmakers, and there is likely to be a lively debate in Brussels on whether — or more particularly, on what terms — to grant a second extension. Britain was originally scheduled to leave the bloc on March 29, but European leaders granted a short extension to give Parliament more time to approve the withdrawal deal.

Mrs. May and Mr. Corbyn met on Wednesday, and teams from both sides continued the discussions on Thursday. The session ended with neither breakthroughs nor breakdowns.

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