Mrs May is planning on offering MPs a “bold offer” in the hope it will be voted through Commons next month and before she steps down as Prime Minister.
The move follows the final collapse of cross-party talks with Labour aimed at finding an agreed way forward which would allow Britain to leave the EU with a deal.
The WAB – which is needed to ratify the deal with Brussels – is expected to include new measures on protecting workers’ rights, an issue where agreement with Labour was said to have been close.
However, Government sources made clear the package would not just be aimed at Labour MPs but would seek to secure the widest possible support across the Commons.
It is expected to include provisions on future customs arrangements with the EU and on Northern Ireland, including the use of technology to avoid the need for border controls with the Republic.
It will not, however, seek to re-open the Withdrawal Agreement – which included the controversial Northern Ireland “backstop” – after the EU repeatedly made clear it could not be re-negotiated.
Mrs May, writing in the Sunday Telegraph, said: “I still believe there is a majority in Parliament to be won for leaving with a deal.
“When the Withdrawal Agreement Bill comes before MPs, it will represent a new, bold offer to MPs across the House of Commons, with an improved package of measures that I believe can win new support.
“Whatever the outcome of any votes, I will not be simply asking MPs to think again. Instead I will ask them to look at a new and improved deal with fresh pairs of eyes – and to give it their support.”
Mrs May has said she will bring the WAB before MPs for its second reading vote in the first week of June following the short Whitsun recess.
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.
The Prime Minister expected to set out details of her WAB proposals in a major speech before the end of the month.
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.
Another defeat would almost certainly see a ratcheting up of demands for her to go immediately, amid intense frustration at her failure to deliver on the 2016 referendum result.
Nigel Evans, the executive secretary of the 1922 Committee, said: “You can watch the movie Titanic a hundred times, but I’m afraid the ship sinks every time.
“An increasing number of Conservative MPs – even those who voted for it a second or third time – are saying enough is enough.”
Mrs May’s new plan comes amid demands that no deal preparations are ramped up.
Brexit Secretary Steve Barclay insisted no deal contingency plans will be restarted “at pace” if Theresa May’s deal is defeated in the Commons.
He warned that if MPs don’t back the existing deal Britain will be on course to leave without one in five months’ time.
And he hit out at Remainers who assume they’ll be able to delay Brexit yet again if the deal falls next month.
I do think we need in those circumstances to bring forward our preparations to mitigate no deal, because we will need to use the additional time we have
Mr Barclay told Sky News: “Members of Parliament do need to face facts, and if the deal were not to go through then there are only two alternatives.
“Any idea of a second referendum is just a proxy for revoke. You either leave with a no deal, or you revoke.
“If Parliament won’t back a deal, then it needs to confront that reality.
“I do think we need in those circumstances to bring forward our preparations to mitigate no deal, because we will need to use the additional time we have, and we need to move at pace to do so.”
Mr Barclay praised the Prime Minister’s efforts to deliver Brexit, saying: “She’s sacrificed her premiership in order to fight for a deal.”
International Development Secretary Rory Stewart said the Conservatives and Labour are “half an inch apart” on Brexit.
Appearing on BBC 1’s Andrew Marr Show, Mr Stewart said: “We do in many ways agree; none of us wants to remain in the EU, none of us wants a no deal Brexit, which means logically there has to be a deal.
“If there is to be a deal, the Labour and Conservative positions are about half an inch apart,” he declared.
Meanwhile, Labour leader Jeremy Corbyn gave his strongest backing yet for a second referendum.
The Labour leader insisted a second Brexit referendum would not be “disastrous” but said there should be something different for the public to vote on.
Mr Corbyn has been criticised by Labour members and MPs who say Labour should come off the fence and back a second poll.
Asked if a second referendum would be disastrous Mr Corbyn told BBC One’s The Andrew Marr Show: “No, I don’t think anything like that is disastrous but I think it has to be an opportunity for public debate and public discussion, but it has to be about something and that’s why I have made the point clear about a customs union and trade and rights protection.”