But under the government's plans the United Kingdom would be free to diverge from EU rules over services, a major part of the British economy, with ministers acknowledging this will reduce the levels of access available to European markets.
After a marathon 12-hour meeting at her Chequers residence on Friday, the prime minister seemed to have persuaded the most vocal Brexit campaigners in her cabinet to back plans for a new UK-EU agreement.
But he warned: 'We've now got to see the EU's reaction and everyone has got to stay disciplined and keep to the collective responsibility or else we just carry on going around in circles'.
He stressed that the European Court of Justice's "remit in the United Kingdom will end when we leave" but "of course there will be a degree of looking from one group of judges to another".
But critics - including business leaders and MPs - say the plan would be "unworkable" and could cost the Conservatives the next election.
"But we'll do it in a way that protects.and enhances our economy for the future", May said. So all of these structures, we are outside, that is what people voted for.
Tories who attended the first briefing organised by Downing Street on the plans said their concerns had been met, and Mrs May will hope that restive Brexiteers also fall in line when they hear more about the proposals which will be set out in a white paper on Thursday. The proposal would allow free movement of goods, but not of services.
Ministers also acknowledged that "it was responsible to continue preparations for a range of potential outcomes, including the possibility of "no deal".
John Longworth, a chairman of campaign group Leave Means Leave, accused May of personally deceiving Brexit campaigners. "May's Brexit means BRINO - "Brexit In Name Only" - a fake Brexit".
Pro-EU Labour lawmaker Chuka Ummuna described it as "yet another behind-closed-doors stitch up that would leave us all worse off".
Although there had been concerns about the knock-on effect such a position would have on the UK's ability to strike trade deals with the likes of the United States, there were enough "soothing words in there for Leavers", one source said.
As she held the crisis talks with her ministers, the chief executive of European planemaker Airbus, Tom Enders, accused the government of having "no clue or at least consensus on how to execute Brexit without severe harm".
Friday's meeting at the 16th-century manor house 65km northwest of London came with just nine months to go until the United Kingdom leaves the bloc, and with the European Union warning that time is running out to seal a divorce deal.
He did not comment directly on her new plan, which had not yet been detailed in full, but suggested it may fall short.
May was cautious on whether she will win the support of the EU, saying only that she had "been talking to European leaders over the last week or so".
"But this is good, we have come today, following our detailed discussions, to a positive future for the UK".
The Prime Minister added: "We have also agreed a new business-friendly customs model with freedom to strike new trade deals around the world".
Foreign Secretary Boris Johnson is understood to have backed the plans despite saying it would leave the United Kingdom as a "vassal state" and be a "serious inhibitor of free trade", according to BBC political correspondent Nick Eardley.
They would involve the United Kingdom paying "due regard" to European Court of Justice rulings relating to the rules Britain will share with Brussels, potentially softening the Prime Minister's red line on the jurisdiction of judges in Luxembourg.
And the agreed negotiating position also hands a big role for parliament to decide whether Britain should continue to follow European Union rules and regulations, recognizing that any rejection of them "would have consequences".
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