She plans to ask the House of Commons to vote in early June on a withdrawal agreement bill, in what May called a "last chance" to seal a Brexit deal.
"We are making a new offer to find common ground in parliament", she said.
The new proposal also places a legal obligation on the government to seek alternative arrangements to the controversial "backstop" arrangement, an insurance policy created to avoid a hard border on the island of Ireland, by December 2020.
Mrs Leadsom, one of the Cabinet's leading Brexiteers, said she would support the Bill, but warned that could change if its provisions were watered down.
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.
May said that the vote on whether to have a second referendum would be conditional on her Brexit bill passing the first voting stage in Parliament.
When comparing a no-deal Brexit, the Brexit deal and remaining in the European Union, 53% of people would support staying.
Sir Nicholas responded: "I think that's a really stupid bovine thing to do".
"Nothing I've heard leads me to believe it's fundamentally any different to the previous bill that's been put forward, so as of now we're not supporting it", he told BBC television.
Corbyn's comments come after weeks of cross-party political wrangling between senior Labour and government MPs collapsed on Friday.
"It's nearly like she is setting up her own political version of the last rites", Thornberry told BBC Radio 4's Today programme.
She said: 'MPs will be rightly wary of offers from a Prime Minister who is about to resign and will probably be replaced by a hard-line successor.
In an impassioned speech Tuesday in London, Ms.
Tory Brexiteer Jacob Rees-Mogg has also hinted he could vote against the deal once again.
Regardless of how the vote goes, she will then meet the chairman of the Tory backbench 1922 Committee, Sir Graham Brady, to agree a timetable to elect her successor as party leader, paving the way for her departure from No 10.
"I recognise the genuine and honest strength of feeling across the house on this important issue", May said in a nationally televised address delivered from the offices of a major accounting firm in London.
While she personally opposed another referendum on the terms of Brexit, the PM said she recognised the "genuine and sincere" feelings on the issue in Parliament.
The Times reported that Chancellor of the Exchequer Philip Hammond will use a speech on Tuesday to attack leadership front-runners Johnson and former Brexit Secretary Dominic Raab for being prepared to accept a no-deal divorce.
Many Tory MPs have been calling on Ms.
Mr Johnson responded by saying that the values of the One Nation group had "never been more important".
Looking uncomfortable as she delivered a televised message to voters ahead of the European Parliament elections, May suggested Labour's internal divisions over a second referendum were to blame for the failure of the talks.
A Conservative party spokesman said: "Lord Heseltine has given more than half a century of service to the Conservative Party and his long-standing and sincerely held views on Europe are well understood".
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