By Agence France-Presse
May said that when she brings the Withdrawal Agreement Bill before parliament early next month, it will come with a new package of measures attached that she hopes can command majority support.
"I still believe there is a majority in parliament to be won for leaving with a deal," May wrote in The Sunday Times newspaper.
MPs three times rejected the deal May struck with Brussels, forcing Britain's EU departure date to pushed back from March 29 to April 12 and again to October 31.
The bill is needed to ratify the divorce agreement struck with the European Union.
On Thursday, May agreed to set out a timetable for her departure following the vote in the week beginning June 3, regardless of whether MPs back her deal at the fourth time of asking.
It is thought that she will trigger a contest for the leadership of her governing Conservative Party once the bill either falls or completes all of its stages through parliament.
- 'New and improved' -
"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," she told the newspaper.
"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."
The bill is expected to include new measures on protecting workers' rights, future customs arrangements with the EU, and on the use of technology to avoid the need for border controls between the United Kingdom and the Republic of Ireland, an EU member state and the UK's only land neighbour.
It will not, however, seek to re-open the withdrawal agreement, which Brussels repeatedly insisted could not be re-negotiated, despite many MPs voting it down due to concerns about its so-called "backstop" clauses on Northern Ireland.
May is expected to set out the details of her proposals in a speech later this month.
- May's poll gloom -
Her gambit on Sunday comes ahead of the European Parliament elections, to be held in Britain on Thursday.
The opinion polls make dire reading for the Conservatives, with the newly-formed Brexit Party forecast to win the most seats, ratcheting up the pressure on May.
The latest survey out Sunday put eurosceptic figurehead Nigel Farage's single-issue party way ahead on 34 percent, with the main opposition Labour Party on 20 percent, the pro-EU Liberal Democrats on 15 percent and the Conservatives on 11 percent.
Furthermore, the poll said the Brexit Party had overtaken the Conservatives in general election voting intentions too, with Labour leading on 29 percent, Farage's party on 24 percent and the Conservatives on 22 percent.
Opinium Research conducted an online survey of 2,004 British adults between Tuesday and Thursday for The Observer newspaper.
Health Secretary Matt Hancock said Saturday that whoever replaces May, calling an early general election risked "killing Brexit" and making veteran socialist Labour leader Jeremy Corbyn prime minister.
"A general election before we've delivered Brexit would be a disaster," he told The Daily Telegraph newspaper.
"A general election before that not only risks Jeremy Corbyn, but it risks killing Brexit altogether."
The next Conservative leader is set to be chosen by the centre-right party's members. Former foreign secretary Boris Johnson is the clear front runner, according to a YouGov survey of grassroots Conservatives for The Times newspaper.
Johnson was on 39 percent, ahead of former Brexit secretary Dominic Raab on 13 percent, interior minister Sajid Javid and Environment Secretary Michael Gove on nine percent each and Foreign Secretary Jeremy Hunt on eight percent.