The pair, who have previously hosted spinoff show I'm a Celebrity... Get Me Out of Here Now! together, will be team captains on the programme.
Minute to Win It sees contestants take on a series of short, funny challenges. It already airs in the US on NBC, and the UK version will begin on ITV2 this summer.
Glasgow-born presenter Darren McMullen, who fronts the Australian version, will apparently also be the host in the UK.
"Caroline and Joe made a great double act on I'm A Celeb... and producers couldn't wait to reunite them," a source told The Sun.
Flack is the new presenter of The Xtra Factor alongside former X Factor contestant Olly Murs.
Meanwhile, Swash is allegedly a judge on new BBC talent show Epic Win.