The plan has also caused outrage among eurosceptic members of May's Conservative party, and foreign minister Boris Johnson joined Davis in dramatically quitting this week in protest
The British government published its long-awaited Brexit blueprint on Thursday that it hopes will restart talks with the EU, but its launch was mired in farce after a protest by MPs briefly suspended a sitting of the House of Commons.
The policy paper, which sets out plans for close economic ties with the bloc after Britain leaves the EU in March, had already sparked two ministerial resignations and revived talk of a revolt against May.
When Brexit Secretary Dominic Raab stood up to present the plan in the Commons, MPs loudly complained they had not seen a copy beforehand - prompting the speaker to suspend proceedings for five minutes to allow them to obtain one.
Raab, who was appointed on Monday after his predecessor David Davis quit, then continued his statement, describing the proposal on offer as "innovative."
It suggests a free trade area and "common rule book" with the EU in goods after pressure from businesses to allow cross-border trade to continue as normal.
Britain would still leave the EU single market and customs union and set its own path on the far bigger services sector, hoping to be able to curb EU immigration and strike its own trade deals with third countries.
'Real blow' for city
For the City of London the plan would accept that firms lose their "passporting" rights to operate in the EU, but seeks a hybrid arrangement.
Catherine McGuinness, head of policy for the City of London Corporation, said the proposals represented a "real blow" for finance firms.
The plan has also caused outrage among eurosceptic members of May's Conservative party, and foreign minister Boris Johnson joined Davis in dramatically quitting this week in protest.
Their departures, followed by a clutch of junior aides, destabilized May's government and revived talk of a leadership challenge.
"What we are doing is delivering on the vote of the British people... that's what our proposal does," she told reporters at a Nato summit in Brussels.
The prime minister is also likely to face some opposition in Brussels, where officials have repeatedly warned Britain to lower its expectations about how close ties can be.
"Of course the EU-27 is open to compromise but not one that can undermine the main pillars of the single market," an EU official said on condition of anonymity.
May has briefed leaders including EU President Donald Tusk and German Chancellor Angela Merkel on her plan and reported a positive response, although they are awaiting the details.
Britain does not have long to argue its case - both sides are aiming for a deal by October, to allow time for its ratification by the British and European parliaments.
Failure to agree would see Britain leave the EU without a deal, with the risk of huge economic disruption on both sides of the Channel.
The Financial Times has reported that one contingency plan being examined in case of a no-deal Brexit is using barges to help keep the lights on in Northern Ireland in case there is disruption to electricity imports from the Republic of Ireland.
'Status of colony'
Britain voted for Brexit in June 2016, but May has so far been unable to present a common position to Brussels on what she wants because of deep divisions in her government.
Amid warnings from businesses that continued uncertainty is risking investment and jobs, and fears time is running out, her cabinet finally agreed on a plan last week.
The aim is to keep EU rules on goods to protect complex manufacturing supply chains, using technology to levy its own duties on UK-bound products from outside the bloc, while diverging on services.
"We need to rise to the challenge and grasp the opportunities" of Brexit, said Raab, adding that the policy represented a "balance."
But Johnson, a leading Brexit campaigner, said following EU rules without being able to alter them risked consigning Britain to the "status of colony," and said it looked like a "semi-Brexit."
Other eurosceptics who want a clean break with the bloc are also livid, prompting speculation they may launch a confidence vote against May.
Brexit-backing MPs, including leading eurosceptic Conservative Jacob Rees-Mogg, will also seek to force her hand by submitting amendments to a trade bill being debated in the House of Commons next week.
Rees-Mogg told AFP his aim was only "to help the government stick to some of its earlier promises."