David Davis has predicted that Theresa May will have to “reset” her stance on Brexit because she will not be able to get her Chequers plan through parliament.

The former Brexit secretary said there were signs Brussels was softening its stance on Northern Ireland, but he warned that EU negotiators would “pile on extra requests” in other areas in the coming weeks before agreeing a deal with the UK.

The prime minister’s warning earlier this week that parliament would have to choose between a version of her Chequers plan or no deal was a false choice, Davis told BBC Radio 4’s Today programme.

He said: “What may well happen is that they pile on extra requests, demands, money, migration, you name it ... Angela Merkel said you must be seen to suffer. All these things will come back and we’ll see more and more pressure. And she’ll have a deal which she won’t be able to bring back to the House of Commons because it will be lumbered with loads of other EU demands. So she’s going to have to have something else.”

Davis predicted that May would have to change tack to avoid the prospects of exiting the EU with no deal. “The time when the decision has got to be taken are going to November, when it is really threatening. Everybody is afraid of no deal, then there will be a point at which it will be possible to reset … there will be other deals on the table at that point.”

He added: “The reset is to step back to what Donald Tusk offered in March. He said you can have a free trade agreement.”

Citing EU trade deals with Canada and South Korea, Davis said: “It is possible. Indeed it is almost possible off the shelf.”

He urged May to use the £39bn divorce settlement agreed last year as a bargaining chip to get a better deal than that proposed in the Chequers plan. He said: “We have a negotiating lever. We have 39bn of them, the £39bn contribution we are going to make in the withdrawal agreement. Once we have signed that away our negotiating leverage evaporates.”

He warned that May would be offered only “warm words” at Wednesday’s EU leaders summit in Salzburg. “I wouldn’t expect much from Salzburg, in terms of outcomes,” he said.

On Tuesday, the EU’s chief negotiator, Michel Barnier, said the bloc was ready to improve its proposal on avoiding a hard border in Northern Ireland. The border issue was the “one area where I think the EU is softening”, Davis claimed.

He pointed out that Barnier appeared ready to accept checks away from the border after ruling them out earlier this year. “Those are the very methods you use to cure the problem of the hard border in Northern Ireland itself,” Davis said.

Davis also dismissed the growing clamour for a people’s vote on the final deal. He said: “To announce a second referendum would be to invite the European Union to give us the worst deal possible in order to persuade us to stay in.

“That’s the consequence of a second referendum and is why it is very bad idea. It is completely against the national interest.”