The government’s favoured successor is Dame Antonia, who was on the four-person shortlist for the role when Sir Chris was appointed.
If given the job, she would be the first female head of the Civil Service.
This would help to counter claims of a “boys’ club” in Downing Street responsible for negative briefings to the media about women.
However, on Wednesday evening Lord McDonald, the former head of the Foreign Office, told Channel 4 News that if this was the case then “in my view, the due diligence has some way still to go”.
He called for a “full process” to appoint a new cabinet secretary that “starts from scratch”.
This has been interpreted as a reference to an investigation Dame Antonia faced over allegations about her spending in 2017 when she was the government’s consul-general in New York, and Lord McDonald was her boss.
In response to Lord McDonald’s interview, a Cabinet Office spokesperson said: “There is absolutely no basis for this criticism.
”Antonia Romeo is a highly respected permanent secretary with a 25-year record of excellent public service.
“The allegations all come from a single grievance made some time ago by a former employee. All the allegations were dismissed on the basis there was no case to answer.”
The Cabinet Office said the appointment process would be agreed by the First Civil Service Commissioner, who is currently Baroness Stuart, former Labour MP and Brexit campaigner Gisela Stuart.
Sir Keir has promised to tighten up vetting for senior roles, after the appointments of Lord Mandelson as US ambassador and Lord Doyle as a peer, despite their links to convicted sex offenders.
As cabinet secretary, Sir Chris had been overseeing the government’s release of documents demanded by Parliament about Lord Mandelson’s appointment.
Badenoch had called for Sir Chris to be kept in the role until this process was complete and for a “full new process” to select his replacement.