“These courts are following what we believe is a nonsensical reading of the Foreign Sovereign Immunities Act,” Schumer said Tuesday. “The fact that a foreign government may have aided and abetted terrorism is infuriating to the families if justice is not done.”

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But the measure has provoked controversy across the political spectrum, as politicians worry it could harm the United States’ relationship with Saudi Arabia and establish a precedent that could come back to bite American officials serving overseas.

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Sen. Lindsey O. Graham (R-S.C.) only recently lifted a hold against the legislation, after Schumer and Cornyn made changes to the bill to accommodate his concerns about its impact on U.S. officials. With that agreement in hand, the Senate passed the bill unanimously on a voice vote Tuesday.

The bill now goes to the House, where Speaker Paul D. Ryan (R-Wis.) has not indicated any support for the proposal.

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But the biggest challenge to the bill may come from the White House, which hinted strongly last month that President Obama would veto it over concerns about sovereign immunity. Recent changes to the legislation did not go far enough because they “were not sufficient to prevent the longer-term unintended consequences that we are concerned about,” White House spokesman Josh Earnest told reporters Tuesday.

Cornyn and Schumer have taken pains to stress that their bill is not specifically targeted at Saudi Arabia and that the kingdom has nothing to worry about if government officials were not complicit in funneling money to groups that supported al-Qaeda and the Sept. 11 attackers.

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But Saudi Arabia has long been the focus of speculation concerning the attacks, particularly when it comes to the government’s support for charities and organizations with links to terrorist groups. Many senators have hinted that a classified 28-page section of a 2002 report from the House and Senate intelligence committees could substantiate that link. Cornyn on Tuesday repeated that the release of those pages “may well be instructive” in establishing Saudi responsibility for the terrorist attacks.

Saudi leaders have denied that the government played any role in the attacks. They have been watching the congressional debate nervously, and threatened last month to start selling off U.S. investments if the legislation passes, according to reports.

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Saudi Arabia holds up to $116.8 billion in U.S. Treasury securities, according to figures released by the Treasury Department on Monday, making it the 13th largest holder of U.S. debt. Officials said the release of the data was unrelated to the Sept. 11 bill pending in Congress.

Supporters of the bill were unmoved by Saudi Arabia’s possibly selling their U.S. asserts.

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“Essentially what the Saudis will be doing by selling off their bonds is suffering a financial loss,” Cornyn said. “They’re not going to suffer a huge financial loss in order to try to make a point, I believe, so I don’t consider it credible.”

Saudi Arabian officials strongly object to the idea that plans to withdraw investments constitute a “threat,” arguing it is what any person or country would do to protect their interests.

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“I say you can warn. What has happened is that people are saying we threatened,” Saudi Foreign Minister Adel bin Ahmed Al-Jubeir said on Monday in Geneva following a meeting with Secretary of State of John Kerry. “We said that a law like this is going to cause investor confidence to shrink. And so not just for Saudi Arabia, but for everybody.”

Cornyn and Schumer also said they are not deterred by the threat of a White House veto.

“I would not vote to sustain a veto, and it’s my guess that most of my colleagues would agree with that,” said Schumer. He added that he thinks the Senate can “easily get the two-thirds” vote to override a veto.