In another part of the rant, which has not been made public, Gibson allegedly threatened to burn the house down.
The NAACP, the Rev. Jesse Jackson and anti-domestic violence groups have condemned the actor for his alleged behavior.
The man's voice in the recording sounds distinctively like the Academy Award-winning actor.
The website reported that Grigorieva made the recording because she was afraid Gibson was going to harm her. The 40-year-old Russian singer and Gibson are engaged in a bitter custody dispute over their 8-month-old daughter.
Gibson is also being investigated for possible domestic violence against Grigorieva from earlier this year. The alleged incident happened Jan. 6 in Malibu. The Los Angeles County Sheriff's Department, which is investigating Gibson's case, is not releasing any further details.
Records in the court case, which may include the audio recordings, remain sealed.
When reached by Eyewitness News, Gibson's publicist Alan Nierob declined to comment on the audio recording, but he did comment on reports by the Hollywood Reporter saying that Gibson has been dropped by William Morris Endeavor Entertainment.
"It is way too soon to be discussing agency representation out of respect to Mel's 30+ years relationship with longtime agent Ed Limato," Nierob said in an e-mail.
According to the Hollywood Reporter, earlier this week, a spokesperson from William Morris Endeavor Entertainment informed the Gibson's representatives that he would no longer be represented by the agency.
AP contributed to this report.
