Kemi Badenoch, the controversial equalities minister, has been mooted as a future education secretary to replace Gavin Williamson in the long-awaited cabinet reshuffle.

Rumours about the rearrangement of Boris Johnson’s top team have abounded for months, with plans scrapped on several recent occasions. But during the summer recess they have been revived.

A No 10 source said Williamson was widely perceived as “terrible” in the education brief and Badenoch would be a “very good” replacement.

Williamson’s position has always been one of the most precarious, and the fallout from this year’s A-level results, when the gap between private and state school grades grew to the widest in the modern era, has probably been seen by Badenoch’s allies as an ideal point to tout her own credentials.

She holds a joint role as exchequer secretary in the Treasury, putting her close to the chancellor, Rishi Sunak – whose stock appears to be on the rise among Tory voters but declining in No 10 after a fallout with the prime minister.

Badenoch is also the minister for equalities and helped pioneer the government-commissioned race report, which was criticised by a UN human rights experts who said it tried to “normalise white supremacy”.

Her “anti-woke” credentials were warmly welcomed by Tory MPs when the Times reported she was in line for a cabinet post on Wednesday. They praised her as “supremely talented”, “very well thought of” among the parliamentary party, and someone who had “a bright future ahead of her”.

Several were impressed by her work on the race report with the Commission on Race and Ethnic Disparities, which was keen to point out the disadvantages faced by white working-class boys while saying it was “difficult to blame racism” for some black students’ educational underperformance.

However, her performance in the equalities brief was not universally praised.

A Conservative backbencher said they “wouldn’t be disappointed” if she were to no longer speak for the government on LGBTQ+ issues, adding she had been “really dismissive” of colleagues urging ministers to take a different approach, particularly on the rights of transgender people.

Badenoch’s appointment would also bring into cabinet someone who is not afraid to be openly hostile with the media. She made headlines this year for publicly branding a journalist “creepy and bizarre” for asking questions about Covid vaccines for a story they did not end up publishing.

Some Tories have also tired of reshuffle talk, saying it is an issue repeatedly ignited partly in an attempt by Downing Street to keep rebellious MPs on their toes.

After aborted reshuffles earlier in the summer, they have become bored of suggestions of a cabinet rearrangement in September, or perhaps late November, or maybe not until 2022.

When it does happen, the timing will be viewed as significant, as the earlier it happens, the more Tory MPs think that will signal an earlier general election.