The group said they would reschedule the gigs due to take place in Glasgow on Friday, and at the O2 in London on Monday, Tuesday and Wednesday.
They did not say who had tested positive for coronavirus.
The Last Domino? tour started in September after being postponed by almost a year because of the pandemic.
Frontman Phil Collins, 70, has been performing seated and has not been able to play the drums because of ongoing health issues, including back problems following surgery.