The Prime Minister said she will seek to ratify her Withdrawal Agreement before the European Parliament elections on May 23, but will make "responsible preparations" to take part if that does not prove possible.
Britain is due to leave the European Union at the end of next week, but Mrs May is now seeking to delay Brexit for a second time after her deal was rejected for a third time last week.
In her letter, she wrote: "I am writing therefore to inform the European Council that the United Kingdom is seeking a further extension to the period provided under Article 50(3) of the Treaty on European Union, including as applied by Article 106a of the Euratom Treaty.
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"The United Kingdom proposes that this period should end on 30 June 2019. If the parties are able to ratify before this date, the Government proposes that the period should be terminated early.
"The Government will want to agree a timetable for ratification that allows the United Kingdom to withdraw from the European Union before 23 May 2019 and therefore cancel the European Parliament elections, but will continue to make responsible preparations to hold the elections should this not prove possible."
Mrs May said if ongoing talks with Labour do not lead to a "single unified approach soon" then the Government would instead look to establish a "consensus" on options on a future relationship that could be put to the Commons.
She wrote: "The Government stands ready to abide by the decision of the House, if the Opposition will commit to doing the same."
It comes amid reports that EU officials are considering offering the PM a flexible extension - or "flextension" - to the Article 50 timetable.
Mr Tusk is preparing to put the option to EU leaders at a crunch summit next Wednesday in a bid to prevent the UK crashing out of the bloc on April 12, according to the BBC.
Here is Theresa May's letter to Donald Tusk in full:
In the European Council Decision of 22 March, taken in agreement with the United Kingdom, the European Union and the United Kingdom agreed that if the House of Commons had approved the Withdrawal Agreement by 29 March 2019, the period provided for by Article 50 of the Treaty on European Union would be extended until 22 May 2019 to provide for ratification.
The House of Commons declined to approve the Withdrawal Agreement and take up that option. Therefore unless we agree a further extension at the European Council you have convened for 10 April, the United Kingdom will leave the European Union without a deal at 2300 BST on 12 April 2019.
The Government's policy has always been and remains to leave the European Union in an orderly way, and without undue delay.
The House of Commons has not thus far approved the deal that would enable this, nor - despite considerable efforts by both Members of Parliament and by the Government - has it yet found a majority in favour of any other proposal. The House has, however, continued to express its opposition to leaving the European Union without a deal. The Government agrees that leaving with a deal is the best outcome.
This impasse cannot be allowed to continue. In the UK 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.
That is why the Government has decided to take further action to seek a consensus across the House of Commons on the right way forward.
I therefore met the Leader of the Opposition earlier this week to discuss whether we might be able to agree a proposal that can be put before the House of Commons which allows the United Kingdom to leave the European Union with a deal. We agreed follow-up discussions that are now taking place.
I have also extended an open invitation to Members of Parliament more broadly to work with me to achieve a consensus that respects the result of the 2016 referendum. I am clear that all of these discussions need to be based on acceptance of the Withdrawal Agreement without reopening it, as the United Kingdom agreed with the European Council at our last meeting, and should focus on the framework for the future relationship between the United Kingdom and the European Union. If a consensus is going to be found, compromise will be needed on all sides, in the national interest.
If the talks do not lead to a single unified approach soon, the Government would instead look to establish a consensus on a small number of clear options on the future relationship that could be put to the House in a series of votes to determine which course to pursue. The Government stands ready to abide by the decision of the House, if the Opposition will commit to doing the same.
These steps demonstrate that the Government is determined to bring this process to a resolution quickly.
The Government acknowledges, however, that after approval to the Withdrawal Agreement is achieved, the process of enacting those commitments in domestic law and therefore ratifying the Agreement in the United Kingdom will take time. Therefore having reluctantly sought an extension to the Article 50 period last month, the Government must now do so again.
It remains the Government's view that, despite this request to extend the Article 50 period, it is in the interests of neither the United Kingdom as a departing Member State, nor the European Union as a whole, that the United Kingdom holds elections to the European Parliament.
However, the United Kingdom 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.
The Government is therefore undertaking the lawful and responsible preparations for this contingency, including by making the Order that sets the date of the poll.
The process I have laid out in paragraphs 4 and 5 is designed to bring the House of Commons to rapid approval of the Withdrawal Agreement and a shared vision for the future relationship, and to allow the Government to introduce the Withdrawal Agreement Bill and so ratify the Agreement.
An important part of that process will be the Government agreeing with the Opposition a programme for the Bill. The Government's objective is to ensure that this programme means the Bill can complete its passage such that the Agreement can be brought into force and the United Kingdom withdraw from the European Union in time to cancel the European Parliament elections.
I am writing therefore to inform the European Council that the United Kingdom is seeking a further extension to the period provided under Article 50(3) of the Treaty on European Union, including as applied by Article 106a of the Euratom Treaty. The United Kingdom proposes that this period should end on 30 June 2019.
If the parties are able to ratify before this date, the Government proposes that the period should be terminated early.
The Government will want to agree a timetable for ratification that allows the United Kingdom to withdraw from the European Union before 23 May 2019 and therefore cancel the European Parliament elections, but will continue to make responsible preparations to hold the elections should this not prove possible.
It is frustrating that we have not yet brought this process to a successful and orderly conclusion.
The United Kingdom Government remains strongly committed to doing so, and will continue to act as a constructive and responsible Member State of the European Union in accordance with the duty of sincere cooperation throughout this unique period. I would be grateful for the opportunity to update our colleagues on the position at our meeting on Wednesday.