Senators unanimously approved the Justice Against Sponsors of Terrorism Act which now heads to the House of Representatives, where Speaker Paul Ryan has voiced reservations.
The White House has repeatedly stated that President Barack Obama, who visited the kingdom in April in a bid to soothe strained ties, opposes the law because it would essentially waive the doctrine of sovereign immunity.
"This legislation would change long-standing, international law regarding sovereign immunity. And the president of the United States continues to harbor serious concerns that this legislation would make the United States vulnerable in other court systems around the world," White House spokesman Josh Earnest said.
He explained that the United States was more engaged internationally "than any other country in the world," particularly in peacekeeping and humanitarian operations.
Undermining immunity, he said, would put Americans working abroad at risk.
The bill would allow 9/11 families to pursue cases in federal court against foreign governments, notably Saudi Arabia, and demand compensation if such governments are proven to bear some responsibility for the attacks.
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Fifteen of the 19 hijackers on 9/11 were Saudi citizens. But no official Saudi complicity in the Al-Qaeda attacks has been proven, and the kingdom has never been formally implicated.
In February Zacarias Moussaoui, dubbed the 20th hijacker, told US lawyers that members of the Saudi royal family donated millions of dollars to Al-Qaeda in the 1990s.
The Saudi Embassy denied Moussaoui's claims. But his accusations revived debate over whether the Obama administration should release a still-classified 28-page section of the 9/11 Commission Report.
The New York Times reported last month that Saudi Foreign Minister Adel al-Jubeir told lawmakers in Washington that the kingdom would be forced to sell up to $750 billion in treasury securities and other assets in the United States to avoid having them frozen by federal courts.
House Speaker Ryan, a Republican, has expressed hesitation about bringing the bill to the floor for a vote.
"I think we need to review it to make sure that, that we're not making mistakes with our allies," he told reporters in April.
The two Democratic presidential candidates, former secretary of state Hillary Clinton and Senator Bernie Sanders, have expressed support for the bill, whose main sponsors are Democrat Chuck Schumer and his Republican colleague John Cornyn.