Update: David Cameron has said he does not regret calling the Brexit referendum which led to the vote to leave the EU.
The former British prime minister told the BBC: "I do not regret calling the referendum.
"It was a promise I made two years before the 2015 general election, it was included in a manifesto, it was legislated for in Parliament.
"Obviously I regret that we lost that referendum. I deeply regret that.
"I was leading the campaign to stay in the European Union and obviously I regret the difficulties and problems we have been having in trying to implement the result of that referendum."
He added: "I support the Prime Minister, I support her aim to have a partnership deal with Europe. That what needs to be put in place. That is what Parliament needs to try to deliver now."
Update: European Commission spokesman Margaritis Schinas said that president Jean-Claude Juncker was available for further discussions with Theresa May in the coming days.
But he noted that the EU27 had made clear the Withdrawal Agreement is "not open for renegotiation".
Mr Schinas told a Brussels press conference: "We are always ready to meet and to talk.
Asked if Mr Juncker would be in Brussels for possible meetings with Mrs May on Thursday or Friday, Mr Schinas said: "You can see that he is here, he is available."
Mr Schinas said no request had been received from London for an extension of the two-year Article 50 negotiation process, under which the UK is due to leave on March 29.
He said: "There has been no UK request for an extension. Should there be a UK request to extend, setting out the reasons for such an extension, it will be a matter for the EU27 to decide by unanimity."
Mr Schinas made clear the Commission believes that the ball is now in the UK's court.
"The next move has to come from London," he told the Brussels press conference. "There is nothing else we can do from here at this stage.
"What matters at this stage ... is that we know what to expect from the UK, and that we don't know."
After Theresa May's defeat the British Parliament remains fundamentally split over what form of Brexit would be best for Britain.
The British Prime Minister is fighting to retain her grip on power while attempting to find a Brexit compromise that could command majority support in the British House of Commons.
Former Ukip leader Nigel Farage has called on the Prime Minister to quit, while there are increasing calls for a second referendum, especially from MPs within the UK's Labour Party.
Mrs May faces a vote of no confidence today after suffering a massive parliamentary defeat over her controversial EU withdrawal agenda.
Labour launched a bid to oust the British Government after Mrs May’s Withdrawal Agreement was overwhelmingly rejected by 432 votes to 202 last night.
Nigel Farage told ITV1's Good Morning Britain: "If the Prime Minister had any sense of honour she would resign this morning having led us to this debacle, but of course she won't."
Describing Mrs May's Withdrawal Agreement as being like "a surrender document of a nation that had been beaten in war", Mr Farage added: "Brexit is not the problem, this Prime Minister is, she needs to go."
He said he now expected the two-year Article 50 process leading to Brexit to be extended beyond March 29.
Responding to a tweet from EU Commission president Donald Tusk about "the only positive solution", Labour MP David Lammy tweeted: "A People's Vote with the option to remain in the EU."
A People's Vote with the option to remain in the EU. https://t.co/XoCxjc6hgC— David Lammy (@DavidLammy) January 16, 2019
Labour's Shadow chancellor John McDonnell said there had been no approach from Downing Street to Jeremy Corbyn to enter discussions on the way forward on Brexit.
After Number 10 made clear that Theresa May remained committed to securing an "independent trade policy" after Britain leaves the EU, he warned she was in danger of "negating" any talks before they started.
"She is now laying conditions down about those discussions which look as though they will prevent any discussion of a permanent customs union," he told BBC Radio 4's Today programme.
"That is what most of the other opposition parties support so she seems to be negating the discussions before they have even started."
Around 45 Labour MPs gathered outside Parliament to voice their support for a second referendum on Brexit.
They held a banner reading: "Labour MPs, MEPs, conference and members agree - The people should have the final say."
Labour MP Chuka Umunna, a leading voice among calls for a second referendum, has shared a statement from 71 Labour MPs who want to put the decision on Brexit back to the British people.
He tweeted: "I'm one of 71 MPs who have just issued this statement: 'we must join trade unions, our members and a majority of our constituents by then unequivocally backing the only logical option to help our country move forward: putting the decision back to the people for a final say'".
I’m one of 71 MPs who have just issued this statement: “we must join Trade Unions, our members and a majority of our constituents by then unequivocally backing the only logical option to help our country move forward: putting the decision back to the people for a final say” https://t.co/3OAE15w7ku— Chuka Umunna (@ChukaUmunna) January 16, 2019
Conservative MP James Cleverley dismissed the statement, however, tweeting: "And now we see the first attempt to ignore the 2016 referendum result and prevent Brexit.
"All these Labour MPs stood on a manifesto commitment to respect the result."
Leader of the Commons Andrea Leadsom said that if Mrs May won the vote of confidence, she would seek to hold discussions with senior parliamentarians on the way forward.
"The Prime Minister will then not necessarily be looking for new ideas that no-one has thought of before, but actually seeking a consensus, a fresh initiative to find a solution that is negotiable with the European Union and that would command a majority in the House of Commons," she told BBC Radio 4's Today programme.
"The Prime Minister has been clear that she wants to talk to senior parliamentarians about what would command a majority and that is what we will be doing over the coming days."
Mrs Leadsom acknowledged that there had been no contact with Jeremy Corbyn but said the Labour leader needed to come forward with constructive proposals.
"I don't think she has written him off at all but he needs to come to the table and tell us what he wants to do," she said.
"He has not put forward any specific constructive proposal and that is a problem, which is why the Prime Minister will be engaging right across the House with those who do have very sincerely held views but want to constructively deliver on what the vast majority of parliamentarians voted for."