Tory MP Jacob Rees-Mogg has called for a fundamental change in ministers' tone on Brexit, accusing UK negotiators of being "cowed by the EU".
The Eurosceptic backbencher said in a speech that "close alignment" with the EU after Brexit would be unacceptable.
Chancellor Philip Hammond, meanwhile, has said he hopes the UK and EU economies will only move "very modestly" apart after Brexit.
He said they were already "completely interconnected and aligned".
BBC political editor Laura Kuenssberg said the war of words went right to the heart of Mrs May's struggle to keep her party together over its biggest project - how the UK will leave the EU.
While the majority of ministers thought she was the only person able to do it, the mood in the party had turned sour in recent weeks with one senior Conservative MP saying they were "in the mood for regime change".
The UK is due to leave the European Union in March 2019, and negotiations are taking place between the two sides.
One of the key questions is how close their trading relationship will be once the UK has left.
Mr Rees-Mogg, one of the leading Eurosceptic voices on the Conservative benches, has recently become the chairman of the European Research Group of Tory MPs.
In his speech, he warned against Brexit being treated like a "damage limitation exercise".
People "did not vote for the management of decline", he said.
"They voted for hope and opportunity and politicians must now deliver it."
1. Trouble ahead- Rees Mogg, who represents many Tory backbenchers will say tonight, govts tone on Brexit 'needs to fundamentally change' - 'approach seems to be we must accept what the EU will allow us to do...this is no way to negotiate and no way for this country to behave— Laura Kuenssberg (@bbclaurak) January 25, 2018
2. Backbench Brexiteers getting their say before David Davis makes his speech on transition tomorrow - think what you like about Rees Mogg but as head of the ERG group he has a lot of sway, and ministers know he is popular with Tory grassroots— Laura Kuenssberg (@bbclaurak) January 25, 2018
The North East Somerset MP said some of the "really obvious opportunities" to improve people's lives from Brexit were at risk, if a model similar to the EU's single market and customs union was adopted.
This would leave the UK "divested of even the limited influence we currently have".
Mr Rees-Mogg also said businesses will suffer unless the UK can set its own regulations, independent of the EU.
And he criticised the UK negotiators, who are led by Brexit Secretary David Davis.
"For too long our negotiators seemed to have been cowed by the EU," he argued.
"Their approach seems to be that we must accept what the EU will allow us to do and build from there. This is no way to negotiate and it is no way for this country to behave."
Mr Rees-Mogg told BBC Radio 5live's Brexitcast that his aim in criticising the government's approach to Brexit was to "support the prime minister against naysayers".
Speaking at the World Economic Forum in Davos, Mr Hammond said the UK was not seeking an "off-the-shelf" model to replace its membership of the EU single market and customs union.
The starting point is a position of "high levels of bilateral trade in goods and services," he said.
"So instead of doing what we're normally doing in the trade negotiations - taking two divergent economies with low levels of trade and trying to bring them closer together to enhance that trade - we are taking two completely interconnected and aligned economies with high levels of trade between them, and selectively, moving them, hopefully very modestly, apart.
"And so we should be confident of reaching something much more ambitious than any free trade agreement has ever achieved."
A Cabinet source told the BBC: "The UK is leaving the EU, the sooner Hammond realises that the better. Very modest changes are not what the 52% voted for."
Eurosceptic Tory backbencher Bernard Jenkin told the BBC: "Either the chancellor's been just a little careless in the ambiguity of his remarks or, rather as we suspect, the Treasury's got rather a different agenda. But I think it would be much easier for the prime minister to do her job if everyone just stuck to her script and I think that is what he should do."
He added that while he didn't want one "maybe [the prime minister] needs another reshuffle in order to give herself more ministers who support her policy" but denied he was suggesting Mr Hammond be sacked.
Mr Hammond tweeted:
I said in #Davos that gov wants to minimise any reduction in access to EU market post Brexit. And it’s a fact our economies are integrated, that’s the baseline from which we leave the single market and customs union – which clearly represents change.1/2— Philip Hammond (@PhilipHammondUK) January 25, 2018
For anyone concerned - I was clear earlier in same speech at #Davos U.K. will cease to be member of EU on 29 March 2019, and after we will be outside Customs Union and the Single Market” 2/2— Philip Hammond (@PhilipHammondUK) January 25, 2018
Former Conservative MP and Scottish secretary Lord Forsyth told the BBC's Question Time that the prime minister "needs to get a grip on the cabinet and the cabinet needs to get behind her".
He added that Mr Hammond seems to be saying "something that's completely at odds with what the prime minister said in her Lancaster House speech."
Downing Street said Theresa May had used major speeches to talk about the "opportunities that Brexit will provide for the country", and that the government was confident of securing these opportunities in the next phase of negotiations.
Asked whether she agreed with Mr Hammond's comments, the spokesman said: "The cabinet has signed up to the vision the PM has set out in her speeches."