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 the spring several scenes were shot on a stretch of the North Yorkshire Moors Railway near Goathland and Tom Cruise was seen in the village of Levisham, though filming in the area had already wrapped before the Covid outbreak.
Other scenes were filmed in a quarry in Derbyshire.
The masked action hero - who is reprising his role as agent Ethan Hunt - waved to members of the public as he filmed scenes at the 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.