Theresa May should stand down as prime minister as early as next month and avoid the “utter disaster” of the Conservatives taking part in the European elections, the former Tory leader Iain Duncan Smith has insisted.

As a slew of opinion polls provided grim reading for the Tories, the senior Brexiteer said dozens of party associations had written to the prime minister making clear they are “not prepared to fight Euro elections” on 23 May.

Mr Duncan Smith’s remarks came after the chancellor Philip Hammond hinted Ms May could remain in office until October – the new deadline for the Brexit negotiations to conclude.

The prime minister has previously made clear she will resign once the first phase – formally cutting ties with the EU – is delivered, and Conservative sources made clear her departure is tied to this, rather than a particular timetable.

But speaking on Sky News’s Sophy Ridge on Sunday, Mr Duncan Smith said: “I know that the prime minister has already said she’s going. She said she would go as and when the agreement was ratified which was looking at around about May, June. I think those dates still stand.

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“I think that what the PM has to do is aim everything now towards departure before the Euros [elections], which would then allow her to step away having done what she said she would do, getting the UK out of the European Union one way or the other, and then we can have another leadership election and pick a new leader which is the way it has to be.”

He blamed a series of concerning polls regarding the party’s prospects at a general election on the delay to Brexit agreed at the EU emergency summit in Brussels.

“It was on the 29th when we didn’t leave that’s when this has all gone wrong. Up until then people were prepared to give Theresa May the benefit of the doubt,” he said. “The big problem was as soon as we didn’t leave you could see all the poll ratings start to crash.”

On participation in the European elections on 23 May – something the UK is legally bound to take part in if a Brexit deal is not agreed by the date of the election – Mr Duncan Smith added: “It would be a disaster for the country. What are you going to say on the doorstep? Vote for me and I’ll be gone in three months. It doesn’t make any sense.

“I always campaign for Conservative candidates but I absolutely do not want to campaign on the Euro elections ticket which would be almost impossible to justify.”

His remarks on Ms May’s leadership of the party came as two former chairmen of the 1922 Committee of backbench Conservative MPs raised the possibility of a change in rules governing challenges to any leader.

Under the current a system, a move against a Tory leader can only be brought once in a 12-month period. It means MPs in the party cannot attempt to topple Ms May again until December – a year on since she successfully saw off a bid by Brexiteers.

Lord Spicer and Lord Hamilton wrote in The Sunday Telegraph: “Conservative MPs are responsible for their party. If they wish to change these rules there is nothing standing in their way.”

Chair of the 1922 Committee Sir Graham Brady also told the newspaper: “It is my understanding that the rules could in future be changed by the agreement of the 1922 executive.”

He added that it was “less certain that it would be possible to change the rules during the current period of grace which was initiated with the triggering of a confidence vote on 12 December last year”.

Speaking on Friday, Mr Hammond said: “The prime minister has said that she will leave once she has done the deal and taken us out of the European Union.