It follows reports that Mr Tusk is prepared to offer the UK a flexible extension - or “flextension” - to the Article 50 timetable of up to a year.
The option will be put to EU leaders at a crunch summit next Wednesday in a bid to prevent the UK crashing out of the bloc on April 12.
In her letter, the Prime Minister writes that the “impasse cannot be allowed to continue” as it is “damaging faith in politics”.
“It is frustrating that we have not yet brought this process to a successful and orderly conclusion,” she says.
“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.”
Mrs May confirms that unless the House of Commons ratifies the Brexit Withdrawal Agreement before EU elections starting on 23 May, the UK will have to vote for new a new set of MEPs.
“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,” she says.
“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.”
Top-level talks aimed at finding a way out of the Brexit deadlock will continue between the Government and Labour on Friday.
Setting out that process as the basis for a further Brexit extension, Mrs May says in her letter to the EU Council President: “If a consensus is going to be found, compromise will be needed on all sides, in the national interest.”