Saturday's live BBC One show would have seen six groups battle for a place on Little Mix's 2021 tour after several weeks of pre-recorded episodes.
It will now air a week later on 24 October - the same day as the first live episode of Strictly Come Dancing's latest series.
Little Mix themselves were not among those who tested positive.
It comes a day after ITV suspended filming on the Britain's Got Talent Christmas special.
A statement from the BBC show's production team said: "We can confirm that a small number of people on the Little Mix The Search production have tested positive for coronavirus and they are now self-isolating following the latest Government guidelines.
"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.
"We hope to be back on air on Saturday 24 October."
The Little Mix show sees the pop group coach and mentor new acts.
The six groups vying to support Little Mix on tour have all been living together and had access to voice coaches, producers and songwriters.
Most of the filming was completed in February, prior to any lockdown restrictions.
Little Mix formed on The X Factor in 2011 and are one of the biggest-selling girl bands in the country.