Ministers will this week discuss sweeteners which could be included in the forthcoming Withdrawal Agreement Bill aimed at securing cross-party support, she indicated.
The cabinet will then consider plans for a series of indicative votes in the House of Commons in a bid to establish which proposals could command a majority.
The new bill – which is needed to ratify the deal with Brussels – is expected to include new measures on protecting worker rights, an issue where agreement with Labour was said to have been close.
Provisions will also be included on future customs arrangements with the EU and on Northern Ireland, including the use of technology to avoid the need for a hard border with the Republic.
It will not, however, seek to remove the controversial Irish backstop arrangement from the bill after the EU repeatedly made clear the issue was not up for renegotiation.
Ms May 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.”
Writing in The Sunday Times, she added: “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."
Ms May has previously said she will bring the new bill for its second reading vote in the first week of June.
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 and prime minister.
But after three previous attempts to get her deal through the Commons all went down to hefty defeats, many Tory MPs are sceptical that her fourth will fare any better.
Another defeat would almost certainly see increased 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, 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."
Shadow Brexit secretary Sir Keir Starmer, who led Labour's negotiating team, meanwhile, was also doubtful that the fresh attempt would succeed.
With Ms May on her way out, he said that a key reason for the failure of the talks was the fear her successor could simply tear up any agreement they reached.
"The Prime Minister said before we started the talks she would be going," he told the BBC Radio 4 Today programme.
"It did mean that during the talks, almost literally as we were sitting in the room talking, cabinet members and wannabe Tory leaders were torpedoing the talks with remarks about not being willing to accept a customs union.
"It put the prime minister in a position where she was too weak to deliver."