Three lawyers filed a motion in January calling for the 66-year-old businessman to be prosecuted over his trip to the Jewish state as Renault-Nissan chairman.
Lebanon is technically still at war with Israel and forbids its citizens from travelling there.
"Prosecutor general Ghassan Oueidat decided... not to prosecute Ghosn for the crimes attributed to him of entering the enemy country and dealing with it economically," the source told AFP.
"A statute of limitations of ten years had passed since the alleged crime," the source added.
Read also: Carlos Ghosn launches executive training programme in Lebanon
Ghosn on January 8 apologised to the Lebanese people for having visited Israel to sign a deal to produce electric vehicles, saying he travelled on business for Renault on a French passport.
He also holds Lebanese and Brazilian nationalities.
The ex-Nissan chief was arrested in Japan in November 2018 on financial misconduct charges and spent 130 days in detention, before he jumped bail and smuggled himself out of the country late last year.
Ghosn appeared at a press conference in Lebanon on January 8, denying all charges and claiming he was a victim of a plot by Nissan and Japanese officials.
Japan has called on Ghosn to return to the Asian country to be tried, while Lebanon has asked Japan to hand over his file on financial misconduct charges.
He and his wife Carole are to take part in a documentary and mini-series about his life, the first of which started shooting in Beirut in September.
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