The actor was one of the stars of the film and claimed Mr Clinton pulled him away from the screening for a private conversation.
In the video, obtained by the gossip website Radar Online, Sizemore says Mr Clinton asked him: "Did you go with Liz Hurley for four years? Do you still see her?"
When Sizemore confirmed the couple had dated but were no longer an item, Mr Clinton is said to have asked for Miss Hurley's phone number. In the recording, Sizemore claims the then-President told him: "Give it to me. You dumb mother******, I'm the Commander-in-Chief of the United States of America. The buck stops here. Give me the damn number."
Mr Clinton then reportedly telephoned her, announcing himself with the words, "Elizabeth, this is your Commander-in-Chief.
"I'm sending a plane to pick you up in three hours. Is that enough time?"
He is said to have sent a private jet to whisk Miss Hurley from Los Angeles to Washington, where she stayed for four days.
Sizemore claimed the affair continued for a year but that "at a certain point (Mr Clinton) told her, 'I don't do love and I'm beginning to think I might love you'."
Responding on social media, Miss Hurley tweeted: "Ludicrously silly stories about me & Bill Clinton. Totally untrue. In the hands of my lawyers. Yawn."
Mr Clinton, whose affair with Monica Lewinsky, an intern, while he was in the White House almost brought down his presidency, has not responded to the reports.