The judicial source told French news agency AFP that the request was rejected because at the time of his arrest an extradition deal between Jordan and France had not entered into force.
The deal was signed in 2011 but became effective only in July last year, after Abassi was released on bail.
Jordan has also refused to hand over a second suspect, Nizar Tawfiq Hamada, because the statute of limitations concerning the criminal allegations against him expired, the source said.
The decision was taken in October, four months after French Foreign Minister Laurent Fabius urged Jordan to "respect international procedure" by extraditing Abbasi and Hamada.
David Pere, a lawyer for the AFVT association that represents French victims of terrorism, said he was "astounded" by the Jordanian decision.
"We were expecting the Jordanian authorities to send a strong message in the fight against terrorism," Pere said.
He said Amman's decision not to extradite the pair was a "political" one aimed at "keeping social peace in Jordan".
The attack on the Chez Jo Goldenberg Jewish restaurant -- in the popular Marais district of Paris -- began around midday on 9 August, 1982 when a grenade was tossed into the dining room.
Two men then entered the restaurant, which had around 50 customers inside, and opened fire with machine guns. They also shot at passers-by as they escaped.
Between three and five men are thought to have taken part in the attack, which was blamed on the militant Palestinian group Fatah Revolutionary Council.
Two other suspects have been named. One is said to be living in Ramallah in the occupied West Bank and the other is a resident of Norway.