The prime minister will feature alongside Steve Rotheram, the mayor of Liverpool City Region, in the charity broadcast.
Commenting on The Celebrity Traitors, Mr Burnham joked that politicians' tactics mean they can spot faithfuls and traitors a mile off.
Mr Burnham defended his appearance on the show, stating he makes no apology for using humour and lightness to engage with the public for a good cause.
A spokesman for Reform UK leader Nigel Farage noted that Farage had previously taken part in a Gogglebox spin-off in 2014 and would happily do so again.
