A group of 60 Conservative MPs are calling on the prime minister to deliver a "hard Brexit".
Former cabinet ministers Michael Gove, Iain Duncan Smith, John Whittingdale and Theresa Villiers have joined the call to pull Britain out of the European single market and the customs union.
They claim getting out of the single market free trade zone is crucial so the UK is not bound by Brussels regulations, while pulling out of the tariff-free customs union is the only way to strike trade deals with countries outside it.
Eleven Labour, DUP and Ukip MPs also reportedly backed the call.
Coinciding with the launch of the European Research Group, a Tory pro-Brexit group, the MPs all backed the statement: "The UK must leave the European Economic Area (EEA) and the Customs Union."
Mr Gove told ITV News it was "really important" to respect the referendum result and "take back control" of Britain's laws and borders and to forge new trade deals.
He added: "That means we need to be fully outside the European Union. Outside the single market and outside what's called the Customs Union."
Mr Gove said there was an important difference between being a member of the single market and having access to it, saying having access meant "being able to trade freely without being bossed about by Euro judges".
Tory MP Steve Baker, who chairs the group, said: "A vote to remain in the EEA or the Customs Union is a vote to be powerless over trade and domestic regulation and therefore poorer than we otherwise can be."
He added: "If we remain in the EEA or Customs Union, we will squander a multi-generational opportunity to relight the torch of liberty and prosperity. I am proud we are standing today against the wasted opportunity and fearfulness which would be to remain within the EU internal market."
It came after a prominent Brexit supporter joined senior Tories in urging Mrs May to scrap an appeal against a court ruling which states that Parliament must approve the triggering of the formal leaving process.