NEW YORK (Reuters) - The White House said on Tuesday it is trying to persuade Congress not to press ahead with a bill that would allow survivors and families of victims of the Sept. 11 attacks to sue the government of Saudi Arabia for damages, in an apparent effort to avoid having a presidential veto overridden.

White House spokesman Josh Earnest, speaking during a press briefing in New York, repeated a promise that President Barack Obama would veto the bill, which has been passed by both houses of Congress, but said there was a push to change legislators' minds about the law.

"White House officials and other senior officials on the president's national security team have engaged members of Congress and their staffs in both parties in both houses," Earnest said.

Congressional aides have said the measure appeared to have enough support, two-thirds majorities in both the Senate and the House of Representatives, for lawmakers to override an Obama veto for the first time since he took office in January 2009.

The Senate sent the bill to Obama last week, giving him a 10-day window, which ends on Sept. 23, to veto the measure.

"We're going to continue that engagement up to and through the president vetoing this legislation in the hopes that we can find an alternative that preserves the effective response that we've already designed to counter state sponsors of terrorism rather than leaving that designation subject to individual decisions by individual judges," Earnest said.

The bill could lead to U.S. diplomats, service members and companies being taken to court all over the world because it would erode the principle of sovereign immunity, he said.

(Reporting by Eric Walsh; Editing by Leslie Adler)