The journalist told her that he had intercepted messages left by Sir Paul to her in 2001, Ms Mills said.
Ms Mills added that after he left the voicemail, the journalist rang her quoting parts of the recording.
When challenged about how they knew what was said, Ms Mills said they admitted the message had been hacked.
The BBC, which conducted the interview with Ms Mills, declined to name the journalist allegedly involved, saying it was not Piers Morgan, editor of the Daily Mirror at the time.
Mr Morgan issued a statement describing Ms Mills’s claims as “unsubstantiated”.