Lapo Elkann
© GC ImagesLapo Elkann
The bad-boy grandson of Fiat founder Gianni Agnelli was busted for allegedly faking his own kidnapping following a two-day bender with a transgender escort at a Manhattan housing project, The Post has learned.

Lapo Elkann, 39, ran up a hefty tab while partying on cocaine, pot and alcohol supplied by the escort — and tried to pay it off by calling a family rep and claiming he'd been kidnapped, law enforcement sources said Monday.

Elkann — who barely survived a 2005 cocaine-and-heroin overdose that involved a transsexual prostitute in Italy — allegedly said he needed $10,000 in cash to secure his freedom from the housing project across from Bellevue Hospital.

But the family rep called the cops, and the escort was busted and charged with grand larceny by extortion after Elkann filed a police report late Saturday, according to the sources.

A subsequent investigation revealed that Elkann had used his own cellphone to call for cash and that he could have left the apartment at any time, the sources said.

The charge against the escort was dropped and detectives hit Elkann on Sunday with a misdemeanor for falsely reporting an incident. He was released pending his arraignment on Jan. 25.

In October 2005, Elkann was found naked, comatose and near death in the Turin apartment of a 54-year-old transsexual streetwalker, after which he left a top job at Fiat and moved to the United States for 18 months.

During that time, he got busted by the NYPD in January 2006, but that case was sealed and details were unavailable, according to law enforcement sources.

Elkann's latest misadventure began on Thanksgiving, when he arrived in New York City by plane and immediately began contacting various online escort services, sources said. He met up with a female escort on West 49th Street and was then introduced by her to the transgender escort, identified as Curtis McKinstry, 29, the sources added.

In an ironic twist, Elkann posted a video to Instagram on Wednesday saying that he was about to embark on a business trip , noting: "I'll be seeing many interesting characters in the creative field and in the entrepreneurial field in the next few days."

Elkann didn't return a message seeking comment.