Mrs May has previously insisted no United Kingdom prime minister could ever agree to such proposals. In a phone call with May on Monday, Irish Prime Minister Leo Varadkar said he would consider a "review mechanism" for the backstop.
May's spokesperson told reporters she had said that "while 95% of the withdrawal agreement had been concluded, on the Northern Ireland backstop there are a number of issues that we still need to work through and these are the most difficult".
The concerns focus on Attorney General Geoffrey Cox's advice on how an exit from the backstop could be found to make sure the United Kingdom is not permanently kept within a customs union, which would severely curtail its ability to strike worldwide free trade deals - a key prize for Brexiteers.
Mrs May says in the letter that the European Union is still pushing for a "backstop to the backstop".More news: GUBERNATORIAL: Governors races feature fight for party control in states
There was no agreement at a Cabinet meeting on Tuesday as ministers were still discussing a fix for the Irish border issue.
The Prime Minister relies on the support of the DUP's 10 MPs for her Commons majority, votes which may become crucial as she attempts to get a deal through Parliament.
"The Government will not agree anything that brings about a hard border on the island of Ireland". Last night a Downing Street spokesman said: 'The Prime Minister's letter sets out her commitment.to never accepting any circumstances in which the United Kingdom is divided into two customs territories'.More news: Prince Louis Adorably Reaches for Granddad Charles in New Photo
Brexit talks had been deadlocked over the so-called backstop - the fallback position that would be activated if a future EU-UK trade deal does not produce a solution that would prevent a hard border between Ireland and Northern Ireland.
Downing Street has played down suggestions that a Brexit deal is imminent, after European Council president Donald Tusk appeared to indicate a breakthrough could come within the next week.
Meanwhile, a senior government source said that any reports in the European media that a deal could come in the next few days should be taken "with a very large pinch of salt".
"This is a negotiation and needs to be an agreement of course between the British Government, but also with the European Union and the 27 countries that are represented by Michel Barnier and his negotiating team".More news: Nadal surgery postpones clash with Djokovic in Saudi Arabia