U.K. Prime Minister Theresa May plans to bring forth "a new, bold" Brexit deal to Parliament in June in an attempt to garner cross-party support after talks with the Labour Party fell through.

The revised deal would provide for a British departure that is "good for jobs, good for our security, and which sets the whole U.K. on course for a bright future outside the EU," May wrote in a column for The Sunday Times. The cabinet will discuss details of the proposed changes in the week of May 20, and consider the possibility of holding indicative votes in Parliament to gauge support for certain changes, she added.

"I will not be simply asking [Members of Parliament] to think again," May wrote. "Instead I will ask them to look at a new and improved deal with fresh pairs of eyes — and to give it their support."

Meanwhile, May is holding discussions with the Democratic Unionist Party in Northern Ireland on potential solutions to secure their support for a deal, the Financial Times reported. The DUP, which props up May's government, is against the planned backstop solution to avoid a hard border between Northern Ireland, which is a part of the U.K., and the Republic of Ireland, which is part of the EU.

The U.K. parliament has already rejected earlier versions of May's Brexit deal three times. If Parliament again fails to agree on a deal, the U.K. will remain on track to crash out of the EU without a deal Oct. 31.

May has already agreed to set out a timetable for her departure as the Conservative party's leader after the fourth vote on her Brexit deal in June, potentially paving way for a new prime minister to be elected and renegotiate a withdrawal agreement with the EU. However, EU leaders have maintained that the deal is not open for renegotiation.