Donald Tusk has said the way to 'dispel the fears and doubts of all citizens concerned' is to initiate Article 50
The British Prime Minister Theresa May says she was right not to guarantee EU citizens the right to stay in the UK after Brexit or it would have left British expats "high and dry".
Mrs May has been accused of using the rights of EU citizens living in the UK as "bargaining chips" but she told MPs she had been right to do so.
However, Mrs May said she wanted to make sure a deal that would give security to Britons in Europe and EU citizens in the UK was struck early on in Brexit negotiations.
She had attempted to strike a deal on the issue ahead of the formal negotiations to put an end to the uncertainty of the 3.9 million EU citizens in the UK and the 1.2 million Brits abroad.
But her advances were rejected by German Chancellor Angela Merkel.
President of the European Council Donald Tusk on Tuesday wrote a letter to 80 MPs, mainly Conservatives, blaming British voters for the "anxiety and uncertainty" felt by British and EU citizens living abroad.
In response to a question on the letter in the British House of Commons, Mrs May said: "I hope that this is an issue that we can look at, at an early stage of the negotiations. Of course there will be two years of negotiations.
"I think it is right to want to give reassurance to British citizens living in the EU and to EU citizens living here in the UK.
"But I think that the reaction that we have seen shows why it was absolutely right for us not to do what the Labour Party wanted us to do, which was simply to give away the guarantee or rights of EU citizens here in the UK, because we have seen that would have left UK citizens in Europe high and dry."
In his letter to MPs, Mr Tusk said: "Would you not agree that the only source of anxiety and uncertainty is rather the decision on Brexit?
"And that the only way to dispel the fears and doubts of all citizens concerned is the quickest possible start of the negotiations based on Article 50 of the treaty?"
During her weekly question session, Mrs May was also introduced to the phrase Smexit by the Conservative MP Claire Perry, to mean a smooth and sharp Brexit.
It comes as the National Institute of Economic and Social Research (NIESR) warned that ending free movement of low-skilled workers from the EU could see businesses close, food prices rise and social care cut.
NIESR's Jonathan Portes told the House of Lords EU Home Affairs Sub Committee agricultural industries may simply go out of business without low-skilled workers.
He said there was no evidence additional immigration lead to reduced employment of UK workers and because unemployment was so low it was "unlikely" employers would react by hiring more British people.
He also said that trying to register the 3.9 million EU citizens in the UK on the date of the referendum would be a "formidable" task and that there was no way of knowing who was in the country on 23 June.