WASHINGTON (Reuters) - The U.S. Senate will vote on Wednesday on whether to override President Barack Obama's veto of a bill allowing relatives of victims in the Sept. 11, 2001 attacks to sue Saudi Arabia.

The vote, which Senate Republican leader Mitch McConnell set as the chamber reconvened on Monday, would be the first action in an attempt by lawmakers to override Obama's Sept. 23 veto of the Justice Against Sponsors of Terrorism Act.

A successful override requires support from two-thirds of lawmakers in both the Senate and House of Representatives, which are controlled by Republicans.

Known as JASTA, the legislation passed the Senate and House in reaction to long-running suspicions, denied by Saudi Arabia, that hijackers of the four U.S. jetliners that attacked the United States in 2001 were backed by the Saudi government.

Fifteen of the 19 hijackers were Saudi nationals.

In vetoing the bill, Obama warned that other countries could use the law as an excuse to sue U.S. diplomats, members of the military or companies, even for actions of foreign organizations that had received U.S. aid, equipment or training.

McConnell said Wednesday's vote will follow two hours of debate divided between Republicans and Democrats. No time for the vote has been set.

(Reporting by David Morgan; Editing by Sandra Maler and Dan Grebler)