About 27,000 recorded messages have been sent out by Twentieth Century Fox to push the DVD and video release of the Steven Spielberg film starring Tom Cruise.
It is believed to be the first time voicemail has been used as a direct marketing device in the UK.
One person has complained to the Advertising Standards Authority (ASA) after receiving a phone message with a clip from the film bearing Cruise's voice.
An apparently breathless Cruise asks, "Where is my Minority Report?", followed by a voiceover urging listeners not to miss the film.
An ASA spokeswoman said a member of the public had complained that the message was "inappropriate and offensive".
"The ASA council will have a look at the complaint to decide whether it is just," she said.
A Fox spokeswoman said all of the people contacted were identified from a database of names used in one of the company's previous promotions.
She said each of them had agreed to receive further information on Fox productions, and had chosen to include their phone numbers.
The company had received no complaints about the promotion.
Direct marketing expert Charlie McKelvey told BBC News Online it seemed like a "desperate" measure by Fox to promote the film.
Mr McKelvey, editor of Precision Marketing magazine, said: "The last thing you want when you pick up your phone is to hear Tom Cruise heavy breathing down the phone at you.
"Marketers will use any channel they can. Many people will have signed up unwittingly to this and it will annoy them if they don't know it's coming."
Charlie said many people were already irritated by direct marketing tactics such as spam e-mails and text messaging.
"Now you are going to get junk voicemail as well, and it's a dangerous precedent," he said.