Tin-eared Theresa May today refused furious Tory calls to quit over Brexit .
The Prime Minister faced down demands to leave her job - whether or not she secures a Brexit deal.
Tory activists and MPs are demanding their leader quits before the summer break and leave Brexit to someone else.
But a Downing Street source said she was standing by her "generous and bold offer" to only go if there's a deal.
The senior figure told journalists: "She has been clear that she will stay to the completion of Phase One."
Labour and Tories hold fresh Brexit talks - but warn deal a 'million miles away'
Phase One refers to signing a Brexit Withdrawal Agreement.
Phase Two is the UK's "future relationship" with Brussels, which is meant to be finished by the end of 2020 but could take several years.
Tories want the PM to quit sooner because, Labour and Tories hold fresh Brexit talks - but warn deal a 'million miles away' prospects of a Brexit deal now look remote.
But the No10 source told journalists: "The Prime Minister made a very generous and bold offer to the 1922 Committee [of MPs] a few weeks ago that she would see through Phase One of the Brexit process.
"And she would leave and open it up for new leadership in Phase Two.
"That is the timetable she's working towards. She wants to get Brexit done."
Despite the briefing, sources told The Sun Mrs May was moving closer to setting out her departure timetable.
Local election results in full: Summary of all 248 council polls across England
Tory pressure has mounted on the Prime Minister to quit after she:
- Local election results in full: Summary of all 248 council polls across England in last week's local elections;
- Delayed Brexit until October 31;
- Entered 'softer' compromise talks with Labour;
- Admitted a deal may only come at the end of July.
Today the 1922 Committee, which represents Tory MPs, will meet once again and discuss changing rules to oust her.
Its leader Sir Graham Brady met the PM yesterday and it's claimed she was set a deadline of 4pm to set her exit date.
This evening the powerful Committee's executive will discuss a crucial '12-month rule' in the Tory party.
This rule prevents any binding no confidence vote in Mrs May as leader, until December - one year after she survived the last effort by 200 to 117.
A previous bid to scrap the '12-month' rule was blocked by the Committee's executive last month.
But that vote was narrow, and some Tory members warned they could change their minds and back a rule change.
One Tory MP brutally confronted Theresa May and demanded she resign - during Prime Minister's Questions.
Brexiteer Andrea Jenkyns said: "She's tried her best, nobody could fault or doubt her commitment and sense of duty, but she has failed.
"The public no longer trust her to run Brexit negotiations. Isn't it time to step aside and let someone else lead our country, our party and the Brexit negotiations?"
Meanwhile the Tories are reportedly doing the bare minimum in the campaign for the EU elections on May 23.
Despite fears they'll be wiped out by Nigel Farage's Brexit Party, a senior No10 source refused to say if the Tories will bother producing a manifesto or even hold a launch event.
Instead the source told journalists: "We're already up and running.
"We announced a full slate of candidates and sent out an election address last week."
The source said they were "certain" the PM will be involved in campaigning but refused to say what she will do or when.
The No10 source added: "Our message is there's only one party that can deliver Brexit."
Journalists replied: "Which party?"
Andrea Leadsom announces she's 'seriously considering' running for Tory leader
Jeremy Corbyn's and Theresa May's ministers will hold a second day of Brexit compromise talks from 6pm tonight.
But prospects of a pact between the two sides looked remote after three hours of talks broke up last night with no agreement.
Today a Labour spokesman warned he expected movement in the "next few days" and it was not an "indefinite" process.
He said the government needed to "move significantly", adding: "The talks are not an indefinite process.
"We agreed to join them at the Prime Minister's request to try and find that majority in Parliament behind a sensible deal that could work for the whole country."
He added: "We are looking, I think, to nail down in the next few days whether and how far the government is prepared to move from its failed deal which has been rejected three times in Parliament."
Labour and Tories hold fresh Brexit talks - but warn deal a 'million miles away'
The Prime Minister is thought to have offered a temporary customs union between the UK and EU to 2022.
But that stops well short of Labour's key demand for a permanent customs union.
Asked if Labour could accept a temporary deal, a spokesman said today: "A new customs union with a UK say is an important and crucial building block of a viable alternative.
"And obviously we've explored that issue and how far the government's prepared to move in a lot of detail."
Chief Whip Julian Smith, Chancellor Philip Hammond, deputy PM David Lidington, chief of staff Gavin Barwell and Environment Secretary Michael Gove are set to attend tonight's talks.
For Labour, Shadow Chancellor John McDonnell will join senior figures Keir Starmer, Sue Hayman and Rebecca Long-Bailey.
