The adaptation of the controversial author's Booker Prize-winning novel is struggling to find distribution in India.
A source told the Mumbai Mirror: "Mr Rushdie and a certain section of India's political hierarchy are not exactly the best of friends.
"We've been told that distributors in India want Mr Rushdie's voiceover removed from Midnight's Children before it can be released here.
"Whether Deepa gives in to the pressure tactics or not is to be seen. In the past, though, she preferred to distance herself from radical elements in India rather than let her creative convictions crumble under pressure."
Rushdie also wrote the screenplay for the film, which is Mehta's first project since 2008's Heaven on Earth.
Midnight's Children centres on life after Indian independence and partition in 1947.
The film stars Slumdog Millionaire actor Irrfan Khan and indie regular Nandita Das.
Das has previously worked with filmmaker Mehta in the acclaimed dramas Fire and Earth.
Mehta reportedly planned to name the adaptation Winds of Change.
The film releases worldwide on November 2.