The Prime Minister's Downing Street statement, in which she blamed MPs for failing to implement the result of the 2016 EU referendum and told frustrated voters "I am on your side", was described as a "low blow" by one former Tory minister.
But Jeremy Corbyn was also under fire after pulling out of a meeting between Mrs May and opposition party leaders because party defectors who are now members of the Independent Group turned up.
The behaviour of the Labour leader, who was also heading to Brussels on Thursday to hold talks with EU Brexit negotiator Michel Barnier, was described by one of their number - Chuka Umunna - as "juvenile" at a time of national crisis.
With just eight days before the UK is due to leave the EU, the Prime Minister will make the case for extending the Article 50 withdrawal process to June 30 at a Brussels summit on Thursday.
Ahead of the meeting, European Council president Donald Tusk said a "short" delay should be possible - but only if MPs finally back her deal before the deadline day on March 29.
With fears in Brussels growing that the UK is heading for a no-deal break, he said he would not hesitate to call an emergency summit next week if that proved necessary.
Mrs May formally made the request for an extension to the end of June in a letter to Mr Tusk on Wednesday.
The Prime Minister had previously indicated she would seek a longer delay after her deal went down to a 149-vote defeat in last week's second "meaningful vote".
However, she reportedly backed down after the threat of Cabinet resignations by Brexiteer ministers, who feared it could spell the end of their hopes of leaving the EU.
In Brussels on Wednesday, Mr Tusk said the question of how long an extension the remaining EU 27 were prepared to grant "remains open".
But European Commission president Jean-Claude Juncker warned Britain would have to leave by May 23 if it did not want to hold elections to the European Parliament - which start on that date - something Mrs May said she is determined to avoid.
Amid exasperation at the political deadlock in London, German foreign minister Heiko Maas said that if they were to extend the deadline, "then we would like to know: Why, why, why?"
France's foreign minister, Jean-Yves Le Drian, said Mrs May would need to convince EU leaders that any delay was to "finalise the ratification of the deal already negotiated" and that it "won't be renegotiated".
With Mrs May struggling to build a majority to back her deal when she brings it back to the Commons for a third time, she made an extraordinary attempt to appeal over the heads of MPs to the public to back her.
In a televised address on Wednesday, she blamed MPs for failing to implement the result of the 2016 EU referendum, and told frustrated voters: "I am on your side."
While she said she would "work night and day" to secure support for her deal, her comments infuriated many of those MPs she needs to win over.
Pro-EU Tory Sam Gyimah, who quit as a minister over her deal, described her remarks as "toxic".
"Resorting to the 'blame game' as the PM is doing is a low blow," he said.
"Democracy loses when a PM who has set herself against the House of Commons then blames MPs for doing their job."
Anna Soubry, who quit the Tories to join the breakaway Independent Group, described it as the "most dishonest and divisive statement from any Prime Minister".
Mr Corbyn was heading to Brussels to "express confidence that an alternative to Theresa May's botched deal can be agreed in Parliament", according to Labour.
Speaking ahead of the talks on Thursday, he said: "Theresa May's botched deal has been overwhelmingly rejected twice by Parliament. It should not be brought back for a third time of asking. Her Government is in chaos, and she is arrogantly trying to bully Parliament to vote for the same bad deal.
"After serious talks with senior MPs from across Parliament, I believe it should be possible to agree a deal with the EU that secures a close economic relationship before the European Parliament elections. I look forward to discussing this with European leaders today.
"It's time for MPs to work together, find a consensus that can get through Parliament, be negotiated with the EU in time and bring leave and remain voters together.
"We believe that consensus can be based on our alternative plan, which would provide protection for manufacturing and jobs, guarantee our rights and end the chaos and uncertainty that the Government is inflicting on our country."