Senior Tories are urging MPs to back away from provoking an early leadership contest this summer, after claims that some want to see David Davis take charge.

Some MPs are keen to see Davis replace Theresa May, claiming that the prime minister is so wounded that she cannot continue in office for long and that the party should act after its disastrous election result.

However, senior figures in the party are now saying it would be a mistake to remove May too soon. Calmer heads are urging restraint, warning that it would be hard to engineer the coronation of Davis – and that unleashing a messy leadership contest could be disastrous for the party.

One senior Tory said there was frustration among a small group of MPs and junior ministers that needed to be reined in. “I’m encouraging everyone to go on holiday,” they said. “It cannot be now – there are some who want it to happen before the end of July, but it is not in the interests of the party. We need to go away, have a holiday and address it in the autumn. There are a lot of conversations going on about when she should go, not if she should go.”

Davis has been cited by some MPs as the best caretaker candidate, but several allies of the Brexit secretary said that he was not involved in any planning about the leadership. “The fact is David is just getting on with his job,” said one. “He’s got the most important ministerial job that anyone has had since the second world war.

“Obviously a lot of MPs are coalescing around him because the task of the Tory party is to deliver Brexit and take on Corbyn and he is superbly qualified to do both, but he is not plotting. There are a bunch of junior ministers who are plotting.”

Becoming leader would be a remarkable achievement for Davis, who was frontrunner to seize the Tory crown in 2005 before David Cameron ultimately emerged as the winner. A poorly received speech at that year’s Conservative party conference was seen as the reason his campaign lost momentum.

Overall, however, there is little appetite now among Tory MPs for a quick leadership contest and even less enthusiasm for an election, which most believe the party would lose. Tory whips have been on the alert for any leadership plotting, but are said to have found no evidence of a groundswell of support for the idea of an early contest.

One minister said: “The ball is in [May’s] court. If over the summer she says, ‘I can’t do this, I’m going to go’, then OK, we have a leadership contest. But unless she does that then she stays and her punishment is to see through Brexit. She has to get on with it and show a bit of resolve.”

Allies of the prime minister pointed to her public comments suggesting she had no intention of stepping aside.

She told a Conservative summer party that the Tories must not lick their wounds after the election. She said this week that the party must be “bold, not timid” – taken as a sign that she wanted to stay in power for some time.

There is also a significant group of newer MPs who believe that their generation should be given time to emerge as possible leadership material. “There is a new generation who are coming up with new ideas,” said one. “We may not want the candidates currently on offer.”

Other young MPs in the party are said to be frustrated by May’s inability to free up slots in the cabinet because of her weakened position after the election.

“Junior ministers think that the cabinet has some good people in there and some rubbish – and that the rubbish should be cleared out,” said a senior party figure. “They know she is too weak to sack anyone.”