Mr Rees-Mogg said the prime minister would find it “extraordinarily difficult” to get her plans approved if she does not come forward with major revisions before the end of the year.
His warning comes as he and other members of the Brexit-backing European Research Group (ERG) of Tory MPs are drawing up alternative plans to Ms May’s, which highlight the benefit of a no-deal Brexit.
Amid the uncertainty and a looming no-deal scenario, support for a new referendum on Brexit is growing, with The Independent having launched its own Final Say campaign with a petition that has attracted more than 670,000 signatures.
Speaking in an interview to The Sunday Times, Mr Rees-Mogg argued that the Chequers deal being pursued by Ms May fails to deliver the Brexit he believes Britain voted for.
He said: “If she sticks with Chequers, she will find she has a block of votes against her in the House of Commons.
“Of course the Eurosceptics in parliament are not in a majority on all issues, but we will inevitably be in a majority on some of them and that will make the legislation extraordinarily difficult if it is based on Chequers.”
He also claimed that if Ms May fails to reach agreement soon with the EU, a rising sense of urgency will benefit the Brexiteers. He then added that letting matters run to December would be “very risky”.
The MP argued that Ms May’s EU withdrawal bill had taken months to pass through parliament, so trying to get the legislation required to implement it passed between January and early March means the government “must come forward with a deal that Brexiteers like”.
If she sticks with Chequers, she will find she has a block of votes against her in the House of Commons
Conservative MP Jacob Rees-Mogg
He added: “Otherwise they might find it’s much harder to get through parliament than they think.”
Mr Rees-Mogg urged Ms May to pursue a “Canada-plus” option, which he believes could command a majority in the Commons.
The ERG is expected to publish a policy paper ahead of the Conservative Party’s annual conference in Birmingham next month.
Meanwhile, the government will on Thursday begin publishing its Brexit technical notices setting out the consequences of crashing out of the EU without a deal, with all 84 due to be out before the end of September.
The explanatory documents expected from the Department for Exiting the European Union are designed to inform citizens and businesses how to cope with a no-deal scenario.
Some are thought to be broad in scope, covering issues like financial services, company law and climate change, while others will focus on specific problems including travelling abroad with pets.
The alliance will see the two organisations mobilise activists and media across the country to capture the growing tide of opinion that a further referendum on Brexit must be held.
An exclusive poll carried out by BMG Research last week found that 48 per cent of the public would now back a vote on any deal struck between the UK and the EU – up from 44 per cent only four weeks ago. Just 24 per cent opposed the idea, down three percentage points over the same period.