Mr Menem, born in Argentina to Syrian parents, stands accused of covering up Syrian connections to the bomb plot in order to protect family friends.
A judge last week accused Mr Menem of diverting public funds in order to cover up the identity of the true perpetrators of the Amia bombing and falsely implicating the Buenos Aires police in the attack.
The judiciary in Buenos Aires issued an international warrant a decade ago for the arrest of eight Iranian diplomats and one Lebanese accused of being behind the bombing.
Iran has always denied any involvement.
The former president has immunity from prosecution after winning a seat in the Argentine Senate, the upper chamber of parliament.
However, lawmakers can vote to revoke the immunity of parliamentarians with a request from the Justice Department.
The prosecution’s argument in the original trial was that a former judge, Juan José Galeano, had halted investigations into a “Syrian trail” on orders from Mr Menem.
That investigation involved Syrian-born Alberto Kanoore Edul, who was detained when the authorities discovered that days before the Amia attack he had telephoned Carlos Telledín, a mechanic who owned the truck that carried the explosives.
Edul, whose parents had a personal relationship with the Menem family, was also a suspect because he owned an address book that included the phone number of Moshen Rabbani, at the time the cultural attaché at the Iranian Embassy in Buenos Aires.
Prosecutors accuse Mr Rabbani of masterminding the Amia bombing and continue to seek his extradition. Edul, who died in 2010, denied involvement.
Public prosecutors called for Carlos Menem to be jailed for six years and Mr Galeano for eight years.
Mr Menem is the second former president to be embroiled in the case. Earlier this year a judge indicted former president Cristina Fernández de Kirchner for treason after she was accused of secretly working to clear Iran of responsibility for the bombing as part of a plan to normalise ties and seal a 2013 grains-for-oil deal between the two countries.
A Jewish Argentine special prosecutor in charge of the investigation into Amia centre attack, Alberto Nisman, accused the Iranian government of having planned the bombing and the Hezbollah terrorist group of having carried it out.
Mr Nisman was appointed special prosecutor in charge of the investigation into Amia centre attack in 2004. Two years later he accused the Iranian government of having planned the bombing and the Hezbollah terrorist group of having carried it out.
In early January 2015, Mr Nisman accused the Ms Fernandez and Héctor Timerman, Argentina’s Foreign Minister, of covering up for the Iranian suspects in the case.
Six days later, just hours before he was due to present his findings in front of the Argentine parliament, he was found dead with a handgun near his body and a bullet in his head.
A subsequent investigation found that he had been murdered, rather than committing suicide as had initially been suggested by the authorities.
