The Senate on Tuesday approved legislation allowing families of Sept. 11 victims to sue the government of Saudi Arabia, defying a White House veto threat as well as threats of economic retaliation from Riyadh.

By voice vote, senators approved the Justice Against Sponsors of Terrorism Act. The bipartisan sponsors of the bill, Sens. Charles Schumer, D-N.Y., and John Cornyn, R-Texas, are now calling on the House to follow suit.

“It’s up to the House,” Cornyn said, while insisting the legislation would not damage the U.S.-Saudi relationship despite what he described as “saber-rattling.”

Schumer said any foreign government that aids terrorists who strike the U.S. "will pay a price if it is proven they have done so."

Saudi Arabia’s government has threatened to pull billions of dollars from the U.S. economy if the plan is enacted.

The legislation gives victims' families the right to sue in U.S. court for any role that elements of the Saudi government may have played in the 2001 attacks that killed thousands in New York, the Washington, D.C. area and Pennsylvania.

Relatives of Sept. 11 victims have urged the Obama administration to declassify and release U.S. intelligence that allegedly discusses possible Saudi involvement in the attacks.

In a statement on Tuesday, a group of them applauded the passage in the Senate and said it “reaffirms the commonsense principle that no person, entity or government enjoys blanket immunity from legal responsibility for participation in a terrorist attack that takes lives or causes injury inside the United States of America.”

But while the bill is a bipartisan measure, the Obama administration has threatened to veto. White House Press Secretary Josh Earnest reiterated that message on Tuesday, saying: “It’s difficult to imagine the president signing this legislation.”

Earnest warned of “unintended consequences,” saying the bill would “change longstanding international law regarding sovereign immunity and the president continues to harbor serious concerns this legislation would make the U.S. vulnerable in other court systems around the world.”

Schumer said Tuesday that the White House concerns, though, “don’t stand up.” He even warned the Senate could override a presidential veto.

The House already is planning to move on the bill. A House Judiciary Committee aide told Fox News the panel intends to hold a hearing on the Senate version of the bill in the near future.

Many Senate Democrats backed the bill, including Senate Democratic Leader Harry Reid, putting them at odds with the White House.

Some lawmakers also have voiced reservations.

Sen. John McCain, R-Ariz., the chairman of the Armed Services Committee, warned that the legislation, if passed, would alienate Saudi Arabia and undermine a longstanding yet strained relationship with a critical U.S. ally in the Middle East.

Fox News' Kara Rowland and The Associated Press contributed to this report.