According to several US media outlets, Elkann, the grandson of legendary Fiat founder Gianni Agnelli, tried to make his family believe he had been kidnapped when he ran out of money during a two-day bender of sex and drugs.
The 39-year-old Italian playboy reportedly spent that time with a transgender prostitute, with whom he consumed a cocktail of alcohol, marijuana and cocaine before running low on funds, the reports said.
He reportedly came up with the plan to ask his family for $10,000 in ransom to pay for more drugs, US media said.
The family alerted police, who arrested Elkann after determining his claims to be false.
Elkann was ordered to appear in a New York court in January for falsely reporting an incident before being released, police said, without confirming whether Elkann was found with a prostitute or whether drugs were involved.
Police did not give the specific date for his next court appearance.
Elkann's brother John took over the helm of the Agnelli empire in May 2008 by becoming the head of the company that manages the family's holdings. The Agnelli family controls about a 30 percent stake in Fiat, and 44 percent of the voting rights.
Elkann made headlines in 2005 when he slipped into a coma after a drug overdose – also reportedly in the company of a prostitute.
On Wednesday, Elkann posted a photo of himself on his Instagram account – dressed in a bright red suit, he said he would be traveling “for business and creative projects around the world in the next few days.”
But US media said that upon his arrival in New York, he sought out a prostitute.
Elkann briefly held a job in Fiat's marketing department, but then moved on to other projects. He is one of the founders of the eyewear brand Italia Independent.