Moves to topple Theresa May are gaining momentum with a string of Tories signalling her resignation could be the only way to “move the dial” and ensure her Brexit deal is passed.

Several backbenchers openly said it was time for the prime minister to quit after recent votes exposed irreconcilable divisions in the Conservative Party from the cabinet down.

Others reported that the PM’s lieutenants have engaged with the idea, sounding out MPs over whether they would vote for Ms May’s Brexit plan if she gave a clearer schedule for her departure. Downing Street made no comment.

The Independent reported on Saturday that such an offer from the PM could mean some of the most hardcore Brexiteers would agree to vote for the withdrawal agreement struck with Brussels.

It comes as talks continue in a bid to win the support of both Labour and DUP MPs, with the latter expected to extract extensive extra funding for Northern Ireland in return.

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As speculation around Ms May’s position mounted on Sunday, one source close to a senior Brexiteer told The Independent: “It will move the dial. Whether it will be enough I don’t know, but it’ll move it.”

Dover MP Charlie Elphicke was among those to publicly say there needs to be “a change of leadership” for him to support the deal.

Mr Elphicke told the BBC’s Sunday Politics South East: “What I am clear on is that if we are going to support it, there needs to be a change of negotiating team.

“So I think we need to have a change of leadership, and a new face and a new team to take us forward to the future relationship.”

Asked if his vote was “up for grabs”, Mr Elphicke added: “We’ve got to get the right future relationship sorted, and that also means having the right team to do it.”

Former cabinet minister Esther McVey has said she is now minded to back the deal, but she also suggested the prime minister should consider setting out the timetable for her departure to win over other rebels.

She said recent cabinet rebellions had made “her position very dubious”, before later adding that the prime minister “needs a dignified departure”.

We need a change of leadership, and a new face and a new team to take us forward to the future relationship Charlie Elphicke MP

Among Brexiteers believed to be wavering in their opposition to the deal are former leader Iain Duncan Smith, Bernard Jenkin and the chair of the European Research Group (ERG), Jacob Rees-Mogg.

A Tory linked to the ERG told The Independent on Saturday: “For some people, even in that hardcore group of the ERG that No 10 finds difficult to reach, it does change things.

“She would need to stand at the despatch box and say I’ll be gone by X date.

“Self-interest alone would have some affect. With her quitting on the cards, do you not think Sajid [Javid] and Jeremy [Hunt] will be saying to their mates, ‘that’s what you should be calling for to back the deal’?”

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Others have echoed the call for change, such as former party co-chair Grant Shapps, who said: “The next stage of the negotiations around Britain’s future partnership with the EU will require a complete change of the negotiating team from bottom to top.”

Ex-Brexit secretary David Davis, who has already come on board with the deal, did not name Ms May but also indicated a new start is needed.

He wrote in The Sunday Times: “We need at least 22 negotiating teams to run all the strands of the negotiation in parallel. We should use experienced trade negotiators in the key areas, not civil servants who are learning on the job.

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“Most of all we should put the whole thing under proper political leadership of the relevant cabinet ministers, not have it run in secret by a Whitehall machine.”

One of Ms May’s most ardent Tory critics, Andrew Bridgen, said he was informed by party whips that Ms May was willing to announce her resignation to get the deal through the Commons.

Others also reported that allies of the PM have been exploring with MPs whether her departure might get the deal over the line.

When asked if Ms May might be the sort of person to realise that she was “part of the problem”, chancellor Philip Hammond said on Sunday: “She absolutely is a kind of person who will always do what she thinks is in the best interests of the country.”