Dozens of moderate Tories, including senior cabinet ministers, have signed up to a powerful new party group in an attempt to stop the Conservatives lurching further to the right during the race to replace Theresa May.

Work and pensions secretary Amber Rudd, already a key figure in the search for May’s successor, is one of the leaders of the newly formed One Nation Group, said to comprise 40 MPs desperate to find a candidate committed to blocking a no-deal Brexit.

The move comes as MPs, including some in government, warn that they believe entryism by pro-Brexit supporters at local Conservative associations risks delivering a leader willing to back a hard break with the EU.

And it follows an outcry after Dominic Grieve, the pro-Remain former attorney general, lost a confidence vote held by his local party.

Boris Johnson, Dominic Raab and Michael Gove – who all campaigned to leave the EU – are among the four frontrunners. The fourth is foreign secretary Jeremy Hunt, who campaigned for Remain in the referendum but has since backed taking a hard line in EU negotiations.

“We’re trying to learn the lessons of last time, where our wing was not organised and did not have enough say,” said one insider. “This time we want to have influence to ensure we act as a caucus that can back someone who supports our kind of agenda.”

Facebook Twitter Pinterest Boris Johnson is one of the frontrunners to replace May. Photograph: Leon Neal/PA

Alongside Rudd, fellow cabinet ministers Greg Clark, David Gauke and David Mundell are understood to be backing the group. Also involved are energy minister Claire Perry, health minister Stephen Hammond and rising stars Rory Stewart and Bim Afolami. Rudd will sit on the group’s board, alongside former ministers Damian Green, Nicky Morgan and veteran MP Nicholas Soames.

The group was formed after a series of secret meetings and dinners held by MPs worried that the party was losing its sense of “compassionate Conservatism”. The One Nation Group is aiming to host hustings in a leadership contest. Candidates looking for their support will need to have been clearly against a no-deal Brexit.

Several Tory MPs are comparing the leadership race to the Grand National, as so many names are being discussed. The comparison is not meant to be favourable. “The problem with the Grand National is that there’s a real danger the 80-1 knacker comes home,” said one. Another MP observed: “Quality horses go to Cheltenham. The nags go to Aintree.”

The setting up of the One Nation Group confirms Rudd’s status as a key power broker in the leadership race, but she has not ruled out running herself. Other MPs on the party’s modernising wing want the group to throw its weight behind health secretary Matt Hancock, a close ally of George Osborne who has managed to steer clear of the Brexit debate. He is being touted as the “fresh start” candidate. Some long-term Tory donors, put off by pro-Brexit candidates, are keen to see Hancock in the race.

Raab, who resigned as Brexit secretary at the end of last year, is also regarded as a “change candidate” for MPs alienated by Johnson. Gove is being touted as the best person to hammer out a trade deal with the EU and has earned respect among some MPs for sticking by May. However, he is now disliked by hardline Brexiters furious that he backed May’s deal.

Gavin Williamson, the defence secretary and former chief whip, has emerged as the other key power broker. “Getting Gavin is a big deal,” said one source close to a leadership contender. “He’s the whip. Whoever secures him will have a big chance.” However, Williamson has not ruled out a leadership bid himself.

Justine Greening, the former education secretary, who backs a second referendum, is also considering a bid, and a number of young MPs are weighing up their chances.

Under party rules, MPs will whittle the number of candidates down to two, and party members will then choose between them. There is now a real concern among mainstream MPs that a radicalised membership will simply select the most hardline candidate. “The party will deny that our membership has become more extreme, but it has – I’ve seen it,” said one member of the government. “The danger is if we put up Hunt against Johnson or Raab, they will go for Johnson or Raab.”

These concerns were heightened by the confidence vote against Grieve last Friday evening, at a meeting that saw him subjected to boos and jeers. Yesterday Grieve blamed a former Ukip opponent for orchestrating the vote against him.

Facebook Twitter Pinterest Dominic Grieve lost a vote of confidence brought by his constituency party. Photograph: AFP/Getty

In his well-to-do Buckinghamshire constituency of Beaconsfield, residents were split on his future. Tom, a Tory member in his 70s who was at Friday’s party meeting, said there had been a group of “loudmouthed yobs” present. He said he had joined the Conservatives six months ago “because I could see the danger that the lunatic fringe would displace Dominic Grieve, and it looks as though that could happen”.

He said Grieve was an “extraordinary MP” and it was very sad that “a man like that should potentially be lost to the government or parliament because of blind prejudice by obsessive Brexiteers”.

Alina, 42 and not a Tory supporter, added: “I appreciated the fact that he had a spine and was one of the very few people in parliament who understood how it worked, and he has integrity.” She said he knew the consequences of what he was going to do, “but still did it”.

“I’m not sure I’d vote for him as a Tory, but if he ran as an independent that would be a different story,” she said.

Steve, shopping at Waitrose, called the vote against Grieve “fucking brilliant”. The 67-year-old said he had always voted for Grieve. “But the major parties stood on manifestos that promised to abide by the [EU referendum] result, and he has continued to defy it. He’s not fit for office. I am happy to see the back of him. I went to the Conservative office myself to see if there was anything I could do to make that happen.”