Andrew Mitchell, an ally of Brexit minister David Davis, reportedly told dinner party that prime minister could not carry on

Theresa May will this week attempt to quash speculation about a potential bid to topple her by relaunching her leadership of the Conservative party and pledging to fight on as prime minister.

May will try to reassert her grip over her party before Thursday’s publication of the EU repeal bill, which is likely to face difficulty passing through parliament in the autumn.

However, her attempts to survive as Tory leader are looking increasingly precarious, amid talk among allies of David Davis, the Brexit secretary, of the possibility of replacing her before the party’s autumn conference.

Andrew Mitchell, a former chief whip and a friend of Davis, is reported to have told a dinner of Conservative MPs that the prime minister needed to be replaced.

One MP who was at the dinner said: “Mr Mitchell effectively said [May] was dead in the water. He said she was weak, had lost her authority, couldn’t go on and we needed a new leader. Some of us were very surprised and disagreed with him.”

Let Theresa May stay at No 10 for the summer, top Tories tell MPs Read more

After the account emerged in the Mail on Sunday, Mitchell, a former development secretary, downplayed the remarks but did not explicitly deny having said them. “This is an overheated report of a private dinner conversation,” he said.

Mitchell, who ran Davis’s unsuccessful bid for the party leadership in 2005, is alleged to have made the comments at a dinner on 26 June, after May struck a deal with the DUP to prop up her minority administration.

There were separate reports that allies of Davis were contemplating circulating a letter calling on May to name a date for her departure, but that Mitchell told them to “go and lie down in a darkened room and then take a holiday”.

In a second warning about May’s waning authority, Grant Shapps, a former Conservative chairman, said the prime minister would need to change her leadership style if she wanted to survive another year at the top.

Writing in the Sun on Sunday, Shapps criticised the dysfunctional, arrogant and corrosive attitude of May’s team in No 10 before the election. “During year two, Theresa May will need to operate a completely different model to remain in power,” he said.

“She must throw open Downing Street to welcome innovative ideas, listen to business and make better use of the party’s broad talent in parliament and further afield. “Trusting others and sharing power beyond a tiny Praetorian guard may not be her instinctive approach, but doing so now could still help her go beyond just about managing the year ahead.”

Some junior ministers are believed to be considering a bid to get rid of the prime minister before the party’s autumn conference. However, the majority of Tory MPs are reluctant to back a change in leadership for fear of another general election, and the prospect of a victory for Labour.

David Lidington, the justice secretary, told the BBC’s Andrew Marr Show that such speculation was typical of Westminster summer parties, as “almost every July too much sun and too much warm prosecco leads to gossipy stories in the media”.

He insisted that “the public has had an election and I think they want politicians to go away and deal with the real problems”.

Rumblings of discontent about May’s leadership come in a crucial week for her premiership as the government publishes its EU repeal bill on Thursday. The bill aims to transpose all EU law into British law to replace the European Communities Act, ready for the moment of the UK’s exit.

Both the Conservative and Labour frontbenches support the principle of leaving the EU but May will still face a battle to pass the bill unamended in the autumn.

MPs from all parties fighting against a hard Brexit are planning a series of attempts to change the legislation, with Labour certain to push for more safeguards for the environment and employment rights.

Some senior Conservatives also sense an opportunity to change the prime minister’s course on cutting ties with the European court of justice (ECJ). Nicky Morgan, the former education secretary, has said this position is too absolutist. Ed Vaizey, a former culture minister, argued this weekend that the UK must stay in the Euratom, the body governing cross-EU nuclear cooperation, which is underpinned by the ECJ.

The mood has alarmed some of the most dedicated Brexiters, who are concerned there is a concerted effort underway to “stymie” the Brexit process.

David Jones, a former deputy minister in the Department for Exiting the European Union, wrote in the Mail on Sunday that “fanatically pro-EU MPs, both Labour and misguided Tory ones, too, have hatched a cunning plot” to water-down Brexit by keeping the UK in the single market and customs union.

He said: “Parliament exists to reflect and enact the will of the people – not to subvert it. But amid the heat of high summer, it looks very much as if a plot is under way at Westminster.

“A plot to stymie the clear, unequivocal wish of the British people to leave the EU, as expressed in last year’s referendum. And it is, I very much regret to say, a plot that may yet succeed if MPs who truly respect that referendum result allow this conspiracy to proceed.”