John Humphrys has attacked the BBC for prioritising equal pay for female staff inside the corporation in the face of threatened cuts to the wider reporting output.

“By the time all the pay complaints have been dealt with, there will be even less money left for programmes,” he wrote in his column in the Daily Mail on Saturday, singling out the Woman’s Hour presenter, Jane Garvey, for criticism.

“It still grates to hear presenters such as Garvey behave as though they are making a heroic sacrifice for humanity rather than enjoying a privileged job most people would sell their souls for,” Humphrys complained. Garvey has spoken out in support of calls made for fairer pay inside the BBC following the resignation of the China editor Carrie Gracie two years ago.

The attack prompted an angry response from Catherine Mayer, co-founder of the Women’s Equality party, who has accused Humphrys of “going full troll” since leaving Radio 4’s Today programme last year.

“He was already wildly inappropriate during his final years in the job. Now he is ascribing to Jane Garvey and the other BBC women a position they do not have,” said Mayer.

This weekend fears have also been voiced about the imminent unveiling of a BBC plan to cut costs by about £40m across its news services, with the axe said to be most likely to fall on some flagship shows presented by women, including Radio 4’s The World At One. The programme’s main presenter is Sarah Montague, who has previously received compensation for being paid less than her male colleagues.

“Last year after a long period of stressful negotiations, I accepted a settlement of £400,000 subject to tax and an apology from the BBC for paying me unequally for so many years,” Montague revealed last week.

Suggested cuts to other high-profile BBC news programmes, including Newsnight and the Today programme on BBC Radio 4, have prompted concern that the recent advances made by the corporation’s women journalists, who now front much more of the BBC’s broadcast news, are likely to be wiped out.

BBC Two’s late-night news programme Newsnight is edited by Esme Wren and its key presenting team includes Emma Barnett, Emily Maitlis and Kirsty Wark. The Today programme editor is Sarah Sands, former editor of the Evening Standard, and in the past couple of years editions have often been presented by a duo of women, drawn from a regular team that now includes Mishal Husain and Martha Kearney.

On Thursday, the presenter and interviewer Victoria Derbyshire learned from a report in the Times newspaper that her award-winning show on BBC Two is an intended victim of the upcoming cuts. Fran Unsworth, the BBC’s director of news and current affairs, sent out an email to all staff telling them the dropping of the Derbyshire show was part of a “wider series of changes” the BBC will announce this week.

Mayer argues that as a national broadcaster the BBC has to represent all parts of society and should make sure that women are allowed to shape news coverage without being at a financial disadvantage.

“Unless you think that white men are way more talented than anyone else, and I suspect that John Humphrys does think he is more talented, then the failure to pay women equally is a failure to recognise their talents,” she said.

The changes are being proposed because BBC News has to save £80m as part of an efficiency drive to find £800m to help fund free TV licences for over-75s.

Last week Lord Hall, the director general of the BBC, announced he is to stand down from his post early.