The Prime Minister updated the Commons on the latest developments in the Brexit negotiations, saying the talks were entering the "final stages", comments which prompted laughter from some MPs.
Mrs May added: "This is the time for cool, calm heads to prevail and it's the time for a clear-eyed focus on the few remaining but critical issues that are still to be agreed."
Mrs May addressed MPs on this afternoon after talks over the weekend failed to produce the hoped-for breakthrough on the so-called "backstop" arrangements for Ireland.
The impasse means that meetings between "sherpas" in Brussels have been cancelled, and UK Government ministers will have no withdrawal plan to approve at Cabinet on Tuesday, before Mrs May travels to the European Council summit the following day.
European Commission spokesman Margaritis Schinas said: "Despite intensive negotiations several key issues remain unresolved. I can say that no further negotiations are planned ahead of the European Council."
The PM said there had been a "great deal of inaccurate speculation" about how the talks were progressing.
She told MPs "real progress" had been made on the withdrawal agreement and the political declaration on future relations.
Mrs May said progress had been made on Northern Ireland but two problems remain – the EU's demands over the Northern Ireland backstop and the need for a time limit.
BREXIT: Theresa May was giving the latest update on talks with the EU
Mrs May said she believed a deal was still within reach.
She told MPs: "I continue to believe that a negotiated deal is the best outcome for the UK and for the European Union.
"I continue to believe that such a deal is achievable.
"And that is the spirit in which I will continue to work with our European partners."
HOUSE OF COMMONS: Theresa May was laughed at after insisting progress has been made on Brexit
LABOUR: Jeremy Corbyn accused Theresa May of a 'blindfold Brexit'
Jeremy Corbyn urged Mrs May to "put the country before her party" and stand up to the "reckless voices" on the Conservative benches.
Amid heated scenes in the Commons, the Labour leader said: "For too long this country has been held hostage to those in her party who want to drive through a race to the bottom deal that lowers rights and standards, sells off our national assets to the lowest bidder.
"It is clear that the Prime Minister's failure to stand up to the warring factions of her own side have led to this impasse."
Mr Corbyn said Mrs May's "blindfold Brexit" was a "bridge to nowhere" and a "dangerous leap in the dark".
He added: "The choice for this Parliament should never be her deal or no deal: if this Government cannot get a good deal for this country then it has to make way for those who can.
"The Prime Minister faces a simple and inescapable choice: be buffeted this way and that way by the chaos of her own party, or back a deal that can win the support of Parliament and the people of this country."
JEREMY CORBYN: Labour's leader accused the PM of a 'dangerous leap in the dark'
The PM's official spokesman also said that "real progress" had been made on a number of issues and the Government was determined to press on with talks.
But he said that the EU continued to insist on the possibility of a carve-out for Northern Ireland which could see a customs border down the Irish Sea – something Mrs May has already said would be unacceptable to any British Prime Minister.
Britain is offering a temporary backstop arrangement under which the whole UK would remain in the EU customs area unless an open border in Ireland was secured by a broader trade agreement.
Discussions were said to have broken down after EU negotiators demanded a "backstop to the backstop", under which Northern Ireland would stay in the customs union after the time-limited UK-wide arrangement came to an end.
NO DEAL: Theresa May warned about the ongoing talks but was hopeful for getting a deal
Former foreign secretary Boris Johnson said talks were "entering the moment of crisis" and urged Mrs May to reject the EU's offer and scrap the backstop.
"The EU is treating us with naked contempt … (offering) a choice between the break-up of this country or the subjugation of this country, between separation or submission," Mr Johnson told the Daily Telegraph.
Mrs May's future as Prime Minister is now tied to her success with Brexit, with her approach already triggering resignations and unrest within the Conservative Party.
Mr Johnson quit the Cabient over Mrs May's plans during the summer, and is believed to have ongoing ambitions to unseat her and take over as Prime Minister.