The Paris prosecutor's office considers there to be "sufficient evidence" against Hassan Diab, who has been in preventive custody since his extradition from Canada in 2014, to try him over the October 3, 1980 attack, the sources said.
The explosion, which left four dead and around 40 wounded, was the first fatal attack against the French Jewish community since the Nazi occupation in World War II.
Diab, a 64-year-old Canadian of Lebanese descent who taught sociology at an Ottawa university, is accused of having carried out the attack on behalf of the Special Operations branch of the Popular Front for the Liberation of Palestine (PFLP).
French investigators believe he planted the bomb inside the saddle bag of a motorbike parked outside the packed synagogue near the Champs-Elysees, where hundreds of people had gathered for Sabbath prayers.
By way of evidence, they point to a sketch of the bomber resembling Diab, the discovery of a passport in his name with entry and exit stamps from Spain, where the bomber is believed to have fled, and testimonies that Diab was a member of the PFLP in the early 1980s.
Diab insists that he was taking exams in Beirut at the time of the attack, which witnesses have corroborated.
The prosecution has admitted to "doubts" about his whereabouts but said it is a matter for a court to resolve.
The final decision on whether the case should go to trial will be taken by an investigating magistrate.
Diab was arrested by Canadian police in November 2008, at the request of French authorities, and extradited six years later.
He has been charged with murder, attempted murder and destruction of property as part of a criminal organisation.
On two occasions, he was granted bail only to be taken back into custody after the decisions were overturned on appeal.
A group of Canadian artists, activists and politicians, including filmmakers Atom Egoyan and political activist Naomi Klein, have taken up his case, urging Canadian Prime Minister Justin Trudeau to intervene to secure his release.