The DUP has urged the Prime Minister to seek changes to the withdrawal agreement as she heads to Brussels today to ask for a short Brexit delay.
Mrs May will make a formal request for a three-month extension of Article 50. She is expected to bring her Brexit plan to the House of Commons for a third vote next week.
In an address from Downing Street last night, she blamed MPs for failing to implement the result of the EU referendum and told voters who want Brexit to be over: "I am on your side."
Earlier she met the Westminster leaders of the Opposition parties, including the DUP's Nigel Dodds.
Mr Dodds said: "It was a useful opportunity to be briefed by the Prime Minister. She is well aware of the concerns that we have outlined about the withdrawal agreement.
"We have always indicated that we stand ready to support a sensible deal which works for all parts of the UK. The referendum result must be respected and there must be no attempt to undermine the democratic will of the people."
In her televised message directed at voters, Mrs May said it was a matter of "deep personal regret" that she was asking the EU for a Brexit delay.
MPs - who rejected her deal by 230 votes in January and 149 last week - had been "unable to agree on a way to implement the UK's withdrawal", she stressed.
"Of this, I am absolutely sure - you the public have had enough," the Prime Minister said.
"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.
"You want this stage of the Brexit process to be over and done with. I agree. I am on your side. It is now time for MPs to decide."
The Prime Minister said MPs had so far done "everything possible" to avoid making a decision on the way forward.
She warned that an extended delay would mean staging "bitter and divisive" elections to the European Parliament at a time when the country needed bringing back together.
"I passionately hope MPs will find a way to back the deal I have negotiated with the EU," she said. "I will continue to work night and day to secure the support of my colleagues, the DUP and others for this deal. But I am not prepared to delay Brexit any further than June 30."
Earlier, European Council president Donald Tusk said he believed a short delay "would be possible" after he spoke to the Prime Minister by phone.
But he said an extension, which must be agreed unanimously by the EU27, was likely to be conditional on Mrs May succeeding in forcing her twice rejected Brexit deal through Parliament.
He appeared to acknowledge the difficulties facing her, describing the hopes of success as "frail, even illusory".
He continued: "Although Brexit fatigue is increasingly visible and justified, we cannot give up seeking until the very last moment a positive solution.
"We have reacted with patience and goodwill to numerous turns of events and I am confident that also now we will not lack the same patience and goodwill at this most critical point in this process."
If the delay is approved by EU leaders, Mrs May will rush legislation through both Houses of Parliament next week to remove the March 29 leaving date from Brexit laws.
Labour leader Jeremy Corbyn pulled out of last night's Opposition parties talks with the Prime Minister after members of the breakaway Independent Group turned up at the meeting.
Independent Group spokesman Chuka Umunna branded it "extraordinary behaviour in a national crisis".
And Liberal Democrat leader Sir Vince Cable said: "Jeremy Corbyn's kinder, gentler politics was found wanting this evening as he stomped out of the meeting with the Prime Minister before it began."
Speaking in the Commons earlier, Mr Corbyn accused Mrs May of not offering "the leadership the country needs".
Bringing back her twice rejected deal to Parliament without significant changes while "threatening a no deal outcome ruled out by MPs is unacceptable and reckless", he said.
Alliance deputy leader Stephen Farry said: "After well over 33 months being available to agree and ratify the terms for Brexit, it is simply stunning, reckless and irresponsible that no formal way forward has been agreed with barely over nine days to go.
"Some people, including the Prime Minister, seem willing to gamble the future cohesion and economic prosperity of the UK through a game of chicken, taking this right to the wire.
"But businesses and others have every right to know what their relationship with the EU is going to be in just over 200 hours' time. People's jobs, opportunities and livelihoods are at stake.
"It is inevitable now (that Brexit) must be delayed to avoid that cliff edge. No deal is unthinkable, yet no extension can be taken for granted."