In a warning ahead of Tuesday's Commons showdown on her Withdrawal Agreement, the Prime Minister acknowledged "no one knows" what will happen if her plan is defeated.
The Prime Minister is urging Brussels to give ground in order to help her deal survive Tuesday's crunch vote by agreeing changes to the Northern Ireland backstop measures.
In a speech in Leave-voting Grimsby, the Prime Minister said: "Next week MPs in Westminster face a crucial choice: Whether to back the Brexit deal or to reject it.
She added: "Everyone now wants to get it done, move beyond the arguments, past the bitterness of the debate and out of the EU as a united country ready to make a success of the future."
In a plea to Brussels for support in making changes, she said what the European Union does over the next few days "will have a big impact on the outcome of the vote".
Foreign Secretary Jeremy Hunt warned Brussels that a failure to co-operate on securing changes to the Brexit deal could "poison" relations with the EU for years.
"This is a moment of change in our relationship between the UK and the EU and history will judge both sides very badly if we get this wrong," Mr Hunt told the BBC Radio 4 Today programme.
"We want to remain the best of friends with the EU. That means getting this agreement through in a way that doesn't inject poison into our relations for many years to come."
The EU said technical discussions are ongoing and it insisted Brussels has come forward with ideas to resolve the deadlock.
European Commission spokesman Alex Winterstein told reporters in Brussels: "The EU side has offered ideas on how to give further reassurances regarding the backstop. You are aware of all this."
Labour leader Jeremy Corbyn said the Prime Minsiter's speech was "a sign of desperation" and promised to oppose the deal again when it came to the Commons, instead continuing to push his own proposals for a customs union and access to the single market.
SNP Westminster leader Ian Blackford called on Theresa May to "stop passing the buck and take personal responsibility for the Brexit crisis created as a result of her own intransigence."
Mr Blackford said: "The Tory leader's damaging Brexit deal is fundamentally flawed and would be devastating for Scotland - inflicting lasting harm on jobs, living standards, our public services and the economy.
"With three weeks to go until the UK is due to crash out of the EU, it is vital that MPs reject May's bad deal, extend Article 50, and hold a second referendum on EU membership."