Theresa May has written to the EU seeking a brief delay to Brexit until the end of June, telling MPs that a longer delay would mean “a failure to deliver” on the result of the 2016 referendum.

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Speaking at the start of prime minister’s questions, May said “I am not prepared to delay Brexit any further than 30 June”, implying the possibility that she could step down rather than allow a long delay to article 50.

The prime minister said she would present her Brexit plan to the Commons for a third time, and if it was passed the delay would allow time to implement it. If it was again defeated, parliament would have to decide how to proceed, May said.

As she spoke, No 10 released the text of her letter to Donald Tusk, the European council president, formally seeking the delay, saying only that the time would allow her to ask the Commons again about her plan.

The letter said May had planned to bring her deal back to the Commons this week for a third vote, but did not specify what would happen if the vote was lost.

The announcement prompted angry scenes at PMQs, with Jeremy Corbyn accusing May of having “no plan” and embarking on a reckless course.

“We are now in the midst of a full-scale national crisis,” the Labour leader said. “Incompetence, failure and intransigence from the prime minister and her government have brought us to this point.”

At the start of PMQs, asked by the SNP’s Pete Wishart about her plans, May said she did not want to extend Brexit beyond a point at which the UK would need to hold new European parliamentary elections in May.

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“I do not believe such elections would be in anyone’s interest,” she told the Commons. “The idea that three years after voting to leave the EU the people of this country should be asked to elect a new set of MEPs is, I believe, unacceptable. it would be a failure to deliver on the referendum decision that this house said it would honour.”

She said she had written to Tusk to seek the extension, adding: “The government intends to bring forward proposals for a third meaningful vote.

“If that vote is passed, the extension will give the house time to consider the withdrawal agreement bill. If not, the house will have to decide how to proceed. But as prime minister I am not prepared to delay Brexit any further than 30 June.”

After PMQs, May’s spokesman was asked a number of times if this meant she would resign rather than allow a longer delay, which he declined to comment on.

Corbyn told May her plan had been rejected decisively twice, and she was “ploughing on, with an unachievable, unsupported deal”. He said: “Surely after two big rejections by the house she must have noticed there isn’t much support for the deal that she negotiated?”

He asked the prime minister: “She told the house the EU would only allow an extension to article 50 if there was a clear purpose. She’s travelling to Brussels tomorrow morning to meet the EU leaders. What is her clear purpose?”

May did not respond, but instead accused MPs of self-indulgently delaying the Brexit process, prompting shouts of protest in the Commons.

“The outcome of a long extension would be endless hours and days of this house carrying on contemplating its navel on Europe and failing to address the issues that matter to our constituents,” she said.

“This house has indulged itself on Europe for too long. it’s time for this house to determine that it will deliver on Brexit for the British people. That’s what the British people deserve – they deserve better than this house has given them so far.”

Other MPs also expressed alarm at May’s course of action. Yvette Cooper, the Labour MP who chairs the home affairs committee, said the prime minister’s only plan was asking people to back “the same plan that she has put to this house twice, and that has been roundly defeated twice”.

Cooper said: “Why will she not just open up, just think again, just allow the indicative votes that others have put forward? Because what she is doing, by sticking to this failed plan, is deeply dangerous for our country. In the national interest, I beg this prime minister to think again.”