Mr Williamson has been heavily criticised for his handling of chaotic changes to the exams system during the Covid crisis and is widely expected to lose the role at the next cabinet reshuffle.
Kemi Badenoch, the equalities minister, has been being lined up to take up the key role of education secretary at the next opportunity, according to a report in The Times citing “multiple sources”.
Meanwhile, the Labour leader Sir Keir Starmer has also called on the PM to sack Mr Williamson for failing children. He told the Guardian: "The question the government has to answer is: why was the attainment gap so big before? Why is it even bigger now? Some [schools] were testing very often and some not very often. It led to the widening and now yawning gap between private and state schools."
Despite widespread frustration with Mr Williamson’s performance among Tory MPs, Mr Williamson is said to be telling his colleagues he “knows where the bodies are” in a desperate bid to keep his job.
The education secretary – who ran Mr Johnson’s 2019 leadership campaign – has been under fire since last year’s A-levels fiasco when the government used an algorithm to “moderate” the results, only to perform a messy U-turn.
Ms Badenoch raised eyebrows last year when she criticised “certain people” calling for changes to the history curriculum in British schools in the wake of the Black Lives Matter movement.
The Tory MP said that some campaigners wanted the history curriculum to be taught “in a way that suggests good people [are] black people” and “bad people [are] white people”.
The equalities minister also came under scrutiny over her decision earlier this year to publish journalist Nadine White’s emails on social media in a stream of tweets, accusing her of “creepy and bizarre” behaviour.
However, according to a report in The Telegraph, the vaccines minister Nadhim Zahawi has also been earmarked as a possible replacement as education secretary.
Talk of a change at the top of the Department for Education comes as the government considers ways to overhaul A-levels amid concerns “grade inflation” is undermining confidence in the qualifications.
For the second year in a row, students were given grades based on assessment by their teachers after exams were scrapped, with almost 45 per cent of pupils awarded an A or A* grades.
Mr Williamson said the government would be “looking at different measures” to tackle grade inflation. The government is thought to be looking at replacing the traditional A to E grades with a numbered system in an effort to shore up the credibility of the qualification.
Influential Tory MP Robert Halfon, chair of the education select committee, called for A-levels to be scrapped and replaced with the International Baccalaureate – saying grade inflation had become “baked” into the current system.
However, Labour’s shadow education secretary Kate Green said changing the grading system would not ensure students are given the support they need.
“I think that is just tweaking the system to get the government out of a difficult story,” she told the BBC Radio 4 Today programme on Wednesday.
“We shouldn’t be just playing around with the grading system here, we should be thinking about the quality of teaching and learning that children are receiving,” the Labour MP added.
MPs and education leaders also expressed alarm over the widening divide between private schools and state schools when it comes to achieving top A-level results.
The government’s former education recovery boss Kevan Collins has warned that “growing inequality” in education could be a legacy of Covid-19 if no action is taken.
The former catch-up tsar, who resigned in June, told the Today programme: “I don’t believe the recovery will happen naturally and I think, if we don’t do something tangible about it, we will have growing inequality in our education system.”
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