Saturday's broadcast of Little Mix The Search has been postponed as a result.
Its producers said they hoped to be back on air by the following Saturday, 24 October.
They have not revealed who has tested positive or how many, but did confirm that none of the four members of the UK girl band had contracted the virus.
Those who have tested positive "are now self-isolating following the latest government guidelines," a statement said.
"Due to the format of the show we have made the decision to postpone Saturday's programme. There are rigorous protocols in place to manage Covid-19 as the safety of all those involved in the production is paramount."
The news comes a day after the Britain's Got Talent Christmas special postponed its filming because at least three crew members tested positive.
Another BBC TV talent show, Strictly Come Dancing, filmed its launch show on Monday, and will hope to avoid similar problems in the weeks ahead.
One contestant, YouTube star and singer HRVY, tested positive 10 days before the launch show, but was later given the all-clear.