Several Tory MPs including some who are unlikely to be promoted, said the Prime Minister should change his team of ministers before Parliament goes into summer recess on Thursday.
The party is “looking for a reshuffle”, one said.
But Mr Johnson is not expected to carry one out before MPs leave Westminster for the summer, despite widespread speculation in the spring that a rejig was imminent.
One of the main reasons for making a change would be to remove “ministers that shouldn’t be ministers”, with some MPs fearing a repeat of the exam grades chaos of last summer under UK education secretary Gavin Williamson.
“The government needs sharpening up and some of the ministers need sharpening up,” one MP said.
“There are one or two that certainly should be moved or dropped, definitely.”
Another Tory said: “The education department has not shone in a particularly good light. It’s the one that’s messed up the most [during the pandemic] and hasn’t actually come back with anything productive.”
But they speculated Mr Johnson may hold fire until autumn or even spring next year to use ministers as a firebreak for any more Covid-related disasters.
Exam grades were “a complete fiasco”, the MP said, adding: “But maybe if it’s a complete fiasco again they can get rid of Gavin Williamson after blaming him for the mess.”
The health minister Lord Bethell was also said to be in line for the axe, having been caught up in the scandal that forced the resignation of his boss Matt Hancock as health secretary.
Some sources meanwhile blamed persistent briefings and rumours of a reshuffle for paralysing the UK Government in recent weeks and months.
There were further briefings and rumours around the time of the Hartlepool by-election in May.
Officials would begin preparing packs for new recruits, while filing away difficult decisions they disagreed about with their current minister in the hope their successor would approve them, the source said.
MPs meanwhile see the impact on ministers, with one suggesting some have “given up completely and are just waiting to hear if they are staying, moving or being dumped”.
“Everything awkward is now just stuck in the ‘leave it for next person’ tray,” one said.
Another MP warned Mr Johnson it would be a “mistake” to delay a reshuffle, as appears likely.
“There’s too many people in the wrong jobs, there’s too many people looking to the next job, thinking they are going to move in three weeks’ time,” they said.
Some experienced ministers are also said to want a reshuffle now, fearing they could be “squeezed out”, with a delay that would see more 2017 and 2019 intake MPs come into the frame as they gather experience.
But MPs were sceptical about the chances of any changes next week, with Mr Johnson already having a mini-reshuffle forced on him by the departure from the Cabinet of Mr Hancock, to be replaced by Sajid Javid.
“I think the general attitude before was that Hancock was going to go in this reshuffle,” one of the MPs said.
Downing Street was approached for comment.