She defended her position by saying Britain could not become a "rule taker" when it came to financial services.
"As on security, what I am seeking is a relationship that goes beyond the transactional to one where we support each other's interests".
Tony Blair, the former Labour prime minister who is among the more prominent campaigners trying to halt Brexit, was in Brussels on Thursday, aiming to persuade European Union policymakers to be ready to welcome Britain to remain if it changes its mind - and be ready to show a will to reform, especially to curb the easy migration across the European Union that anxious many Britons.
"But Parliament would always have the final say and if it wanted to change that and have less market access as a result, that is a trade-off that we would be empowered to make because the sovereignty would reside with Parliament". In an attempt to add detail to Britain's negotiation on leaving the EU, May mixed concessions with a plea for a deal that would keep trade flowing between the world's biggest trade bloc and Britain's $2.7 trillion economy.
But Ireland's foreign minister told the BBC on Sunday that the European Union may reject May's plan.
The European Council president Donald Tusk told the United Kingdom prime minister on Thursday that friction in trade was "inevitable" after Brexit. While Michel Barnier, the EU's chief Brexit negotiator, welcomed the "clarity" that May provided, other European Union officials said there were still many questions that urgently need answering.
"So rather than having to bring two different systems closer together, the task will be to manage the relationship once we are two separate legal systems", she will say.
But a spokesman for Mr Johnson said: "This is a yawnerama - a nonsense claim from an anonymous quote about a half-baked and misleading leak that has been totally superseded by the Prime Minister's excellent speech on Friday".
"Most of the points raised were about nuances of language", the source said. They were asked to hand the copies back before leaving.
The EU official called Tusk's lunch with May "an open and honest debate in a good atmosphere about the real political difficulties ahead of us".
But the PM was warned that her vision could take years to negotiate.
The EU's chief negotiator also "welcomed" Mrs May's words.
Mr Coveney said if agreement can not be reached during tri-partite talks between the UK, Ireland and the European Commission, the backstop plan of full British alignment with customs union and single market rules that Mrs May "committed clearly" to in December would have to be put in place.
Mr Barnier urged British politicians to face up to hard choices raised by the decision to leave the EU.
In a major speech on Friday, Mrs May rejected "unacceptable" EU proposals to retain customs union arrangements in Northern Ireland, but accepted the UK's "responsibility" to help maintain a soft border with the Republic - spelling out in detail how she believed this could be achieved by technological means or through a broader trade agreement.
"Abandoning such ideas will enable us to begin building an ambitious future partnership based on the foundation of realism and in the interests of our citizens and our businesses, and this is my objective".