German government sources said he had appealed to senior figures in Angela Merkel's administration to help "make a win-win out of this mess".
"She is too weak to face down the fanatics in her own party and to deliver a final deal that protects jobs and the economy".
The letter from the European Research Group, led by Jacob Rees-Mogg, made clear that Eurosceptics will not tolerate any backsliding.
It is also almost double the number needed to trigger a leadership contest.
It sets out the MPs' "continued, strong backing" for the vision set out in Mrs May's Lancaster House speech in January 2017. "Your government must have the ability to change British laws and rules once we leave, rather than being a 'rule-taker,"' it said.
May has faced criticism from colleagues who say she lacks vision, while Brexit supporters fear she'll sell them short and tie Britain too closely to European Union rules.
"This isn't a letter, it is a ransom note", she said.
After months of internal divisions, May is aiming to win support on Thursday from her inner circle of cabinet ministers for her blueprint for the post-Brexit trade deal between the United Kingdom and the EU.
Sam Lowe, a trade expert at the Centre for European Reform, told the Guardian: "A standstill transition is in everyone's interest and absolutely essential if the government is serious about negotiating a mutually beneficial ongoing relationship between the EU and UK". Euroskeptics see a risk this would lead to the United Kingdom following European Union rules.
In it, the MPs say Mrs May "must" ensure Britain can change its laws without authorisation from Brussels from the moment it leaves the bloc in March 2019.
These officials fear the United Kingdom loses power in the detailed trade talks that will have to take place during the transition phase because the country will already have left the EU.
It also states that the transitional phase-or implementation period-that's now being negotiated must meet their own criteria and be based on World Trade Organization principles.
Responding, Labour MP Chris Leslie said on behalf of the pro-EU group Open Britain: "The ERG have long felt they can dictate Government policy and they are brazenly advocating the hardest of hard Brexits".
The letter from the European Research Group says the United Kingdom should be free to negotiate and sign trade deals with other countries as soon as it leaves.