In a Commons statement on Monday following talks with European leaders in Brussels, the prime minister will insist the “shape of the deal across the vast majority” of the withdrawal agreement is now clear.
But she will also reiterate her refusal to compromise over the Irish border, one of the key issues yet to be resolved with just over five months until Britain leaves the EU.
Ms May’s statement to parliament comes as she faces growing anger among Eurosceptic rebels in her own party as well as calls for a second referendum on the final Brexit deal.
In an attempt to highlight “important progress” since a fractious EU summit in Salzburg last month, the prime minister will tell MPs that agreements have been reached on security, transport and services.
She is expected to confirm that protocols have been developed on how Brexit will impact Gibraltar and the UK’s military base in Cyprus.
“Taking all of this together, 95 per cent of the withdrawal agreement and its protocols are now settled,” Ms May will tell the Commons.
“And all of this from the last three weeks alone, is in addition to the agreements we had already reached.
“The commitment to avoiding a hard border is one that this House emphatically endorsed and enshrined in law in the Withdrawal Act earlier this year.
"As I set out last week, the original backstop proposal from the EU was one we could not accept, as it would mean creating a customs border down the Irish Sea and breaking up the integrity of the UK.
“I do not believe that any UK prime minister could ever accept this. And I certainly will not.”
Furious backbenchers warned the prime minister she is “drinking in the last chance saloon” at the weekend after tensions flared over her negotiating strategy after a Brussels summit.
Senior Brexiteer Theresa Villiers criticised “disturbing” anonymous briefings to Sunday newspapers, including claims the PM was entering the “killing zone”.
But Brexit minister Suella Braverman said her colleagues were free to express themselves in any way they wished and repeatedly refused to say she would back Ms May in a confidence vote.
Brexit secretary Dominic Raab said the exit agreement must be finalised by the end of next month to allow new laws to be put in place in time for exit day.
He suggested a transition extension could run for three months, but said the move would have to solve the Irish backstop issue.
Labour has warned Ms May that it will not back her Brexit blueprint when it reaches the Commons.
Shadow Brexit secretary Sir Keir Starmer said there was a real lack of confidence Ms May could bring back “anything by way of a good deal”.
An estimated 670,000 people marched in London on Saturday to demand a second Brexit referendum.
Conservative MP Anna Soubry said many of her Tory colleagues were privately supportive of a fresh vote amid bitter divisions in the party.