The “electoral consequences could be dire” without a U-turn, because it would be “obvious” to voters that the prime minister’s promises had been broken, the former Brexit secretary said.
The warning comes in an explosive letter sent to all Conservative MPs in which Mr Davis demands a looser Canada-style trade relationship with the EU.
The model was described as nothing more than “a shortcut to a no deal” by Brexit secretary, Dominic Raab, during a heated Commons appearance to set out the state of the talks.
Ms May has also ruled out that option, arguing it would “break up the UK” – by requiring a customs border in the Irish Sea – and because of the damage to manufacturers, which require frictionless trade.
The Liberal Democrats said the letter betrayed the “growing panic rippling through the Conservative party”, as the government’s Brexit strategy faltered.
The EU has since rejected Chequers as well – but the prime minister has refused to officially drop the proposals, ahead of a crucial Brussels summit next week.
In the letter, Mr Davis said the EU would either confirm its rejection of the Chequers plan next week, or demand “further significant concessions”.
But he insisted a “Canada-plus-plus-plus” deal – going further on trade, internal security and foreign policy – was “within our grasp with political will and imagination”.
Mr Davis wrote: “If we stay on our current trajectory, we will go into the next election with the government having delivered none of the benefits of Brexit, with the country reduced to being a rule-taker from Brussels, and having failed to deliver on a number of promises in the manifesto and in the Lancaster House speech.
“This will not be a technicality, it will be very obvious to the electorate. The electoral consequences could be dire.
“So it is in both the party’s interest, and crucially the national interest, that we reset our negotiating strategy immediately and deliver a Brexit that meets the demands of the referendum and the interests of the British people.”
Later on Tuesday, Mr Raab told MPs that talks with Brussels have “intensified” in recent weeks, with both sides are “closing in workable solutions” to all the outstanding issues, including the deadlock over the Irish border.
He stood firm against Brexiteers who urged him to ditch the Chequers plan in favour of their preferred Canada-style model, which he claimed was a “shortcut” to crashing out of the EU without a deal.
However he faced criticism from former Tory home secretary Amber Rudd over the government’s ”gung-ho approach” to no deal, as she warned that security matters are not yet in place to keep the UK safe.
But Mr Davis urged her not to make “preposterous threats” that a refusal by Tory MPs to back her could result in Brexit being lost altogether, or watered down by the need for Labour votes.
Although the EU has offered “Canada-plus-plus-plus”, it has insisted Northern Ireland would then have to stay in the customs union and part of the single market to avoid a hard border – a suggestion that Mr Davis and other pro-Brexit Tories would never accept.
Earlier, Steve Baker, Mr Davis’s former deputy, repeated his warning that up to 80 Tories are ready to vote down the government’s proposals.
Tom Brake, the Liberal Democrat Brexit spokesman, said: “David Davis’s letter highlights the growing panic rippling through the Conservative party that Theresa May’s Brexit is going to cause nothing but misery for the country and the Tory party.
“People should not have to cross their fingers and hope for the Tory party to sort out this terrible mess they have got the country into.
“Instead the people should get the final say on the deal and the option to remain in the EU.”
The Independent has launched its #FinalSay campaign to demand that voters are given a voice on the final Brexit deal.