She says there is "one real sticking point left and a considerable one" which is how to guarantee no return to a hard border in Ireland.
The government has begun formal preparations for a no-deal Brexit, publishing more than 100 technical guidance documents covering many possible impacts.
Those assurances had previously been simply that a political declaration on the future EU-UK relationship, due to be agreed this year alongside the withdrawal treaty that would smooth Britain's March exit, would state that a way would be found to avoid customs posts.
Taking the unusual step of briefing planned remarks to the Commons in advance, May will conclude that "taking all of this together, 95 per cent of the withdrawal agreement and its protocols are now settled" in talks that she has until now largely insisted on keeping secret.
Criticism of her Brexit strategy has intensified following a Brussels summit.
However, during her address Mrs May said: "We would not accept a position in which the United Kingdom, having negotiated in good faith an agreement which prevents a hard Border in Northern Ireland, nonetheless finds itself locked into an alternative, inferior arrangement against our will".
Gibraltar, a British overseas territory located at the tip of the Iberian peninsula and regularly claimed by Spain, will leave the European Union along with the UK.
That prompted May to propose a country-wide alternative in which the whole of the United Kingdom would remain in parts of the customs union after Brexit.
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They also offered to extend the treaty's transition period, when nothing will change, by a year to the end of 2021 to help ensure that a customs deal is ready to avoid a hard border.
The DUP MP was speaking after Brexit Secretary Dominic Raab said the Northern Ireland backstop plan must be dropped if the prime minister wants to extend the UK's 21-month transition period, which is expected to end in December 2020.
"Both the European Union and United Kingdom are in agreement that our future partnership is the answer to this and will provide the solutions in the long term", she said.
Last night a Government spokesman struck a measured tone in response to Mrs May's speech and said "Ireland will listen to proposals that the United Kingdom brings forward, but any proposal has to be legally operative in ensuring that we avoid a Border between Northern Ireland and Ireland".
"That's when we want this to be, so that there is no need for any backstop, or extension of implementation period, or anything else that's been put forward", she said. Her Belfast allies have threatened to vote down her budget next week and there is talk of a party leadership challenge.
"It seems to be getting ever more precarious", he said.
During a heated Cabinet meeting on Tuesday, Lidington said that he was the only person around the table who had been an MP on the fateful day in 1992, when interest rates soared in 24 hours as Britain dropped out of the European exchange rate mechanism.
"A hard Brexit raises the concern that there could be a significant loss of scientists from the United Kingdom, particularly the young scientific talent upon which the country's future will depend", he added.