The blockbuster action sequel, starring Tom Cruise will resume shooting on June 14, according to the studio.
It was unclear how many people tested positive.
In a statement obtained by the PA news agency, Paramount said: “We have temporarily halted production on Mission: Impossible 7 until June 14th, due to positive coronavirus test results during routine testing.
“We are following all safety protocols and will continue to monitor the situation.”
Filming on Mission: Impossible 7 has been taking place around the UK.
In April pictures emerged of a train track set up at a quarry in Derbyshire, said to be attached to the film.
And Cruise, 58, filmed action scenes on top of a moving mock steam locomotive in North Yorkshire.
The masked action hero – who is reprising his role as agent Ethan Hunt – waved to members of the public as he filmed scenes at a heritage railway.
Production of Mission: Impossible 7 has suffered a number of delays due to the coronavirus pandemic.
Filming was scheduled to begin in Italy in February 2020 but production was halted as coronavirus cases rose in the country.
After a delay, filming restarted at locations including Warner Bros Studios Leavesden in Hertfordshire.
In December, Cruise was reportedly recording warning staff on the film they would be fired if they broke on-set coronavirus rules.
Mission: Impossible 7, directed by Christopher McQuarrie and also starring Ving Rhames, Henry Czerny, Simon Pegg, Rebecca Ferguson and Vanessa Kirby, is set for release in May 2022.