The 57-year-old 'Braveheart', who was previously married to Robyn Gibson, is believed to have impressed Givaty with his sense of humor, reported Contactmusic.
"Mel compliments her on her striking beauty. And she loves his sense of humor, they're getting closer by the minute," a source said.
"She thinks he's very charming. They love each other's company," the source added.
Gibson, who had a controversial private life in recent years, was arrested in 2006 for driving under the influence of alcohol and drugs.