The UK is currently due to leave the EU on 12 April and, as yet, no withdrawal deal has been approved by MPs. The prime minister has proposed that if UK MPs approve a deal in time, the UK should be able to leave before European Parliamentary elections on 23 May.
But she said the UK would prepare to field candidates in those elections, in case they do not reach agreement. Meanwhile the BBC's Europe editor Katya Adler has been told by a senior EU source that European Council President Donald Tusk will propose a 12-month "flexible" extension to Brexit, with the option of cutting it short, if the UK Parliament ratifies a deal.
But his proposal would have to be agreed unanimously by EU leaders next week. The prime minister wrote to Mr Tusk to request the extension ahead of Wednesday's meeting.
But she was offered a short delay to 12 April - the date by which the UK must say whether it intends to take part in the European Parliamentary elections - or until 22 May if UK MPs had approved the withdrawal deal negotiated with the EU. They voted it down for a third time last week.
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In her letter, she says the "impasse cannot be allowed to continue", as it was "creating uncertainty and doing damage to faith in politics" in the UK.
She said if cross-party talks with the Labour Party could not establish "a single unified approach" in the UK Parliament - MPs would be asked to vote on a series of options instead which the government "stands ready to abide by".
She wrote that the UK proposed an extension to the process until 30 June and "accepts the European Council's view that if the United Kingdom were still a member state of the European Union on 23 May 2019, it would be under a legal obligation to hold the elections".
To this end, she says the UK is already "undertaking the lawful and responsible preparations for this contingency".
But it said if a withdrawal agreement could be ratified by Parliament before then "the government proposes that the period should be terminated early" so the UK can leave the EU before then, and cancel preparations for the European Parliamentary elections.