The Sun published the expletive-laden audio in which Cruise said: "If I see you doing it again, you're... gone."
The paper said Cruise had seen two crew members "standing too close to one another in front of a computer screen".
Variety and Reuters quoted sources confirming the audio was genuine.
Filming is currently taking place in the UK. The Sun did not say when the incident happened, but film-makers returned to the country in early December, according to Reuters.
The Mission: Impossible franchise is hugely successful at the box office, starring Cruise as Ethan Hunt.
The seventh movie had to pause filming in Italy in February due to concerns about the coronavirus outbreak, and The Sun said the actor had "personally tried to ensure there are no more delays".
In the recording, Cruise can be heard shouting: "They're back there in Hollywood making movies right now because of us. We are creating thousands of jobs.
"That's it. No apologies. You can tell it to the people that are losing their... homes because our industry is shut down.
"We are not shutting this... movie down. Is it understood? If I see it again, you're... gone."
Variety said the film is scheduled for release in November 2021. BBC News has asked for comment from Cruise's representatives and the Mission: Impossible studio and producers.
The audio quickly spread around the internet. Nick Murphy, who directed last year's A Christmas Carol for BBC TV and Save Me for Sky Atlantic, praised the star's actions, writing: "Tom Cruise was right."
US radio host John Rocha also voiced his support, writing: "I wish MORE people in charge would react like this to people who violate protocols or not wearing masks. If only more people saw the bigger picture that Tom is highlighting here."
Dennis Tseng from movie site Collider added: "Tom Cruise ain't wrong. Now he just needs to come back to America and yell at every single anti-masker."
And British actress Rebecca Front joked: "The one thing missing from that #TomCruise audio is the distant sound of a lone drill and an anguished 1st AD [assistant director] shouting 'Can we PLEASE hold the work?!"