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.

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