The Bibles, known as the Crowns of Damascus, were written between 700 and 1000 years ago. They were owned by the Jewish community in Damascus for hundreds of years.
The manuscripts feature ornate decorations of tiny letters (micography), and gold-leaf highlights.
Mossad, the Israeli intelligence agency, took them in the 1990s. They kept their move secret for more than a decade.
Rabbi Avraham Hamra, the leader of the Damascus Jewish community at the time, helped Mossad take the Crowns.
However Rabbi Hamra is now challenging Israel's ownership bid, because of what he says is a broken promise.
He says Israel promised to build a Syrian-Jewish centre in the country, where the Crowns of Damascus would be kept. This centre has still not been built, and given the National Library's official application for ownership, this amounts to Israel breaking a promise, according to Rabbi Hamra.
Israel denies making such a promise.
Hamra also says the Crowns are the "religious and spiritual treasure of the Syrian Jewish community."
A lawyer working for the National Library says it will advertise its bid for official ownership of the Crowns of Damascus in Syria, Europe and the US, so anyone against the proposal can legally challenge it.
The government in Damascus has not commented on the situation.
