Her partner, Allan Little, the BBC's Moscow correspondent, made an overnight dash from Russia to be with Ms McDonald, 44, and her family in the intensive care unit of a central London hospital.
The 39-year-old journalist returned home within hours of hearing of the accident on Friday night in Islington, north London, which left Ms McDonald with serious head injuries.
The Chancellor, Gordon Brown, a former boyfriend of Ms McDonald, was among those awaiting news on her progress as doctors at the University College Hospital in Euston said her condition remained grave.
A hospital spokesman said: "Ms McDonald remains stable but, given the nature of her injuries, critically ill. Her family have been with her all day and remain at her bedside."
Despite the seriousness of her condition, it is understood that there are no immediate plans for the political reporter - a presenter of Channel 4 News, The World This Week and BBC1's On The Record - to undergo surgery.
Mr Little made the dash to London within hours of hearing of the accident. He and Ms McDonald have been together for four years.
A Channel 4 spokesman said: "He left immediately he heard and has been at the hospital since earlier today. Sheena's family, including Allan, continue to be with her." Mr Brown, who was romantically linked with Ms McDonald while they were both at Edinburgh University in the 1970s, and later in 1994, asked to be kept informed of her condition.
The Chancellor, spending the weekend in Scotland with his fiancee, Sarah Macaulay, remains a close friend of the broadcaster.
Reports emerged in 1994 that the pair were again seeing each other before Mr Brown started his relationship with Ms Macaulay, a director of a public relations company, that year.
Newscaster Alex Thomson, who read the Channel 4 News on Saturday night, thanked viewers who had called in with good wishes for the presenter.
The Channel 4 presenter, who chaired the judging panel for the Orange Fiction prize last year, was knocked down at 11.47pm on Friday in St John's Street, Islington.
A Metropolitan Police spokesman said: "As is routine in accidents involving police vehicles, the driver is suspended from driving duties pending a full investigation."
The police van, which had its blue lights flashing and siren on, was answering an emergency call to a fight on nearby Holloway Road.
Scotland Yard said that the driver, a 27-year-old police constable and a 30-year-old colleague, who was a passenger, escaped injury but are suffering shock.
Both are based at Islington police station.Reuse content