By offering the possibility of holding a second vote on her deal and a compromise on customs arrangements, Mrs May hopes to win over opposition Labour MPs, whose votes she needs to overcome resistance in her own Conservative Party.
"I recognise the genuine and honest strength of feeling across the House on this important issue", May said.
Speaking to the BBC this morning, Chancellor Philip Hammond said that "to advocate for no deal is to hijack the result of the referendum and in doing so knowingly to inflict damage on our economy and our living standards".
"Instead I will ask them to look at a new and improved deal with a fresh pair of eyes - and to give it their support".
May warned European Union sceptics with her party ranks that a rejection of her final throw of the dice threatened to boom Brexit for good.
Deputy Lib Dem leader Jo Swinson said she would be open to discussions with the former deputy prime minister if he wanted to join her party.
The current deadline to leave is 31 October.
"Let me remind them: the 2016 Leave campaign was clear that we would leave with a deal".
Mrs May said the Government will commit in law to let Parliament decide on the customs issue and there will be a requirement to vote on whether to hold a second referendum.
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The failure of the talks and May's pledge to set a timetable for her departure increases the risk of a no-deal split from the European Union as Tory MPs jockey to succeed her.
The latest polling shows the British public willing to back the newly formed Brexit Party, which is campaigning exclusively on taking the United Kingdom out of the European Union without a deal, exiting on World Trade Organisation terms.
She said: "Although it's not possible for (alternative arrangements) to replace the backstop in the Withdrawal Agreement, we can start the work now to ensure they are a viable alternative".
Johnson has been one of May's most outspoken critics over Brexit and supports leaving the European Union without a deal.
But she says that will only be the case if MPs are willing to approve the withdrawal agreement bill in the House of Commons in the first week of June.
But after three previous attempts to get her deal through the Commons went down to hefty defeats, many Tory MPs are sceptical that her fourth will fare any better.
The opposition leader, who last week broke off talks with the PM's top team after weeks of deadlock, branded it "effectively a repackaging of the same old bad deal, rejected three times".
The latest survey out Sunday put eurosceptic figurehead Nigel Farage's single-issue party way ahead on 34 per cent, with Labour on 20 per cent, the pro-EU Liberal Democrats on 15 per cent and the Conservatives on 11 per cent.
He said he believed Mrs May was a "decent person" trying to find a middle ground position, but had been thwarted by an "impossible" Conservative Party.