Asked if she took responsibility for the failure to get a Brexit deal through, May said: "No".
"It had been incredibly frustrating that MPs on either side of the Leave-Remain divide had got so sort of entrenched that they just were not willing to make that compromise that would enable us to get the majority to get this through", May said in a BBC interview in 10 Downing Street on Friday.
Ms Miller said the case has been prepared now because she believes a Prime Minister would only prorogue Parliament out of "desperation" and may not have sufficient time after either Tory leadership candidate takes office.
Expanding on his plan, Mr Hunt insisted it was possible to get a revised deal with Brussels by the end of September, and said if it took "a few extra days" to get it through Parliament he would delay Brexit beyond the October 31 deadline.
Sharing screen space, Boris Johnson, Hunt's rival and the frontrunner in the leadership contest, said the United Kingdom would leave by October 31 "come what may", and that if this did not happen it would lead to "a huge erosion of trust in politics".
She added: "I'm hoping that the response we get to the letter is on the advice is given is that, you know, that propagation is would be beyond the power of a Prime Minister so they won't go there".
But pressed on whether the United Kingdom would be out by Christmas, he said: "I'm not going to give you those commitments".
The Foreign Secretary maintained prime ministers should "only make promises they know they can deliver".
Either Mr Hunt or Mr Johnson will take over from Mrs May the next day.
But leading Eurosceptic Tory MP Andrew Bridgen lambasted them, saying: "We need to leave by October 31 and quite honestly, people like Greg Clark and Philip Hammond have been roadblocks to preparing for no deal".
"If you want to leave the European Union quickly, if you want to avoid a general election, which is the risk if you go about this in the wrong way, I'm the person who has the biggest chance of negotiating that deal and getting us out by October 31".
Mr Johnson's lack of explicit support for Mr Darroch during the ITV debate on Tuesday was widely seen to have been the final straw for the envoy following the leak of his diplomatic cables criticising Donald Trump's White House. "There are other things - I think I probably actually should have done the TV debates".
He told the BBC: "I think we've got to come out on October 31 and I think it is very odd that those who are saying they would delay even further can't set another date".
With Parliament not sitting, MPs would not be able to block it - for example, by holding a vote of no confidence in the prime minister.
Former prime minister John Major also said last week he would be prepared to turn to the judiciary if needed to avoid a constitutional crisis.
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