The move comes after a centerpiece of the Getty's antiquities collection was shipped back to Italy after more than a decade on display at the Getty Villa in Pacific Palisades.
The Getty says it purchased the statue of Aphrodite from a Swiss collector, but Italian officials claim it and other items had been looted.
Italy is also fighting to get the "Statue of a Victorious Youth." The 2,300-year-old bronze was discovered undersea by a fisherman in 1964 and made its way into the Getty collection under what some call mysterious circumstances.
Gov. Gian Mario Spacca's plan was described as "an innovative peace treaty in the long-raging battle with the J. Paul Getty Museum over antiquities that Italy claims are rightfully theirs and were illicitly trafficked and sold to the Getty."
"The 'Victorious Youth' is a very important symbol and testimony for the Italian culture," Spacca said through a translator. "We came here not to declare war to the Getty Museum but instead to establish a collaboration."
Getty officials countered by saying the statue rightfully belongs to the museum.
"We were clear at the start of our conversation that the 'Statue of a Victorious Youth,' known as the 'Getty Bronze,' was not a matter for discussion since legal issues regarding this object are ongoing in Italy," said Ron Hartwig, vice president of communications for the Getty Museum. "We continue to be completely confident of our ownership of the 'Getty Bronze' and we will vigorously defend our position."
The Getty said it awaits the rule by the Italian court, and if it loses, it plans to appeal.
City News Service contributed to this report.