Sir Keir Starmer should find the source of leaks about Sue Grey and “publicly fire them”, the former leader of the Scottish Conservatives has said.

Ruth Davidson, co-host of Sky News’ Electoral Dysfunction, told a live event of the podcast that Downing Street’s political advisers need to be “read the riot act” and reminded that “journalists are not your friends”.

During the discussion, former Labour MP turned peer Harriet Harman also suggested the attacks on the prime minister’s chief of staff could be fuelled by sexism.

Baroness Harman said: “There is something about an older woman in authority that some young men find hard to face.”

Ms Gray, whose report in parties at Downing Street during the pandemic contributed to the downfall of Boris Johnson, has found herself at the centre of hostile briefings just as Labour convenes the party faithful for its annual conference in Liverpool – the first one they’ve held while in government for 15 years.

The row erupted on Wednesday when it was reported she received a pay rise after the general election which apparently took her salary above the prime minister’s annual earnings of £167,000.

In an apparent attempt to damage her politically, it was leaked to the BBC that Ms Gray now earns £170,000 and junior staff are disgruntled about it as some of them are being paid less than what they received when Labour was in opposition – despite now occupying more senior government roles.

The government does not deny the salary level but insists she did not set it herself, as some reports claimed.

Ms Davidson said she did not think it mattered if Ms Gray was earning more than the prime minister – and the story is what the leaks say about how Number 10 is operating.

“Sir Keir Starmer should find the source of leaks,” she said.

“For me if I was head of the Number 10 operation right now, I would be pulling in every political adviser and I would read them the riot act.

“I would say journalists are not your friends.. as soon as you are in government they will not be coming to your kids’ christening, your colleagues are who you owe your loyalty to and then I would fire three people publicly.

The event also heard from Home Office minister Jess Phillips, who said she was not convinced by reports Ms Gray could resign over the briefings.

“I don’t think she is going anywhere,” she said, adding: “I don’t recognise the idea of a lack of control or dysfunction in Number 10, from my experience.”