She didn't say when the vote would happen.
Mr Dunne, who has supported the deal in the two previous votes, said he did not want to see a no-deal scenario.
Theresa May has been fighting for her political life after losing key votes to get her Brexit deal through Commons. She needs to win over at least 75 lawmakers to get it through. Instead, she blamed Parliament for the deadlock, and warned that if lawmakers did not back her deal it would cause "irreparable damage to public trust". One lawmaker slammed her remarks as "toxic".
Earlier on Thursday, French President Emmanuel Macron refused to rule out a no-deal Brexit.
He tweeted: "On Brexit, we must be clear to ourselves, to our British friends and to our people".
But her foreign minister Heiko Maas of the junior coalition partner Social Democrats said May's letter "only pushes the solution further down the road". "I sincerely hope that will be with a negotiated deal".
The UK will have to leave the European Union with "no deal" if the British MPs reject the EU-backed accord once again, the French President Emmanuel Macron has warned.
I don't believe that is what you want - and it is not what I want.
Describing the United Kingdom as being in "political crisis", Mr Macron said: "There needs to be a profound political change if there is to be an extension which is anything other than technical".
"That would mean asking you to vote in European Elections, almost three years after our country made a decision to leave", she said in her message.
Some EU states, including Germany, had given a largely positive response to May's well-flagged request.
"You want this stage of the Brexit process to be over and done with this. I'm on your side".
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A short time later, in Hanoi, I visited the dank, miserable cell where McCain spent much of his five-plus years in captivity. McCain says Trump "will never be a great man", and is urging Trump to focus on his family instead of hers.
"You are tired of the infighting, you're tired of the political games and the arcane procedural rows, tired of MPs talking about nothing else but Brexit when you have real concerns about our children's schools, our National Health Service, knife crime".
Many legislators, including some from her own Conservative Party, condemned the tone of her speech.
Anna Soubry, of the breakaway Independent Group, described it in a tweet as the "most dishonest and divisive statement from any Prime Minister". Otherwise, the U.K.is facing a chaotic "no-deal" departure from the bloc within days, or a much longer delay that May says she will not allow while she is in power.
The Prime Minister's Downing Street statement, in which she blamed MPs for failing to implement the result of the 2016 European Union referendum and told frustrated voters "I am on your side", was described as a "low blow" by one former Tory minister.
Elections to the European Parliament are due to begin on May 23. May said it was not in Britain's interests to take part in European elections.
EU officials fear that citizens unable to vote could launch legal action at the European Court of Justice, or that businesses unhappy with legislation adopted by the new parliament might challenge its legitimacy.
Theresa May's decision to formally ask for an extension to Article 50 yesterday was no great surprise, as the motion giving her permission to do this was signed off by the Commons on March 14.
"What kind of message would that send?" she asked.
He said: "I think the Prime Minister has shown remarkable determination and resilience in trying to deliver on the referendum result through the unprecedented challenges of the scale of the task in the face of a minority government, so achieving consensus on any issue is extremely hard".
Britain's opposition leader Jeremy Corbyn was also in Brussels, for talks with the EU's chief negotiator Michel Barnier where they discussed looking for a "constructive alternative" to Mrs May's deal.
Corbyn said: "These are hypotheticals".