A day after the House and Senate overwhelmingly voted to override President Obama's veto, GOP leaders are expressing reservations about legislation that would allow lawsuits related to 9/11 to go forward against Saudi Arabia.

Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellGraham: GOP will confirm Trump's Supreme Court nominee before the election Trump puts Supreme Court fight at center of Ohio rally The Memo: Dems face balancing act on SCOTUS fight MORE (R-Ky.) and House Speaker Paul Ryan Paul Davis RyanKenosha will be a good bellwether in 2020 At indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates MORE (R-Wis.) both said they were open to discussions about changing the bill, which Congress approved unanimously.

"I do think it's worth further discussions, but it was certainly not something that was going to be fixed this week," McConnell told reporters on Thursday.

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McConnell also criticized the lack of a discussion about "the potential consequences" of a very "popular bill."

Ryan agreed that Congress may need to "fix" the legislation but said he wasn't sure when that would happen.

"We want to make sure the 9/11 victims and their families have their day in court," Ryan told reporters. "At the same time, I would like to think that there may be some work to be done to protect our service members overseas from any kind of legal ensnarements that occur, any kind of retribution."

The legislation, known as the Justice Against Sponsors of Terrorism Act (JASTA), became law after the Senate and House voted to override Obama for the first time in his presidency. In the Senate, the vote was 97-1.

The bill was backed by Democratic leaders in Congress and had the powerful support of families of the 9/11 victims.

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Saudi Arabia's government pressed Congress and the administration to oppose the bill, and the White House argued it could have negative repercussions on U.S. citizens and companies who could be subject to suits over U.S. government actions.

Signs that many members of Congress weren't comfortable with the vote emerged almost immediately.

Twenty-eight senators sent a letter to Sens. John Cornyn John CornynTumultuous court battle upends fight for Senate Texas Democrats roll out first wave of planned digital ads as Election Day nears Calls grow for Biden to expand election map in final sprint MORE (R-Texas) and Charles Schumer Chuck SchumerJacobin editor: Primarying Schumer would force him to fight Trump's SCOTUS nominee CNN's Toobin: Democrats are 'wimps' who won't 'have the guts' to add Supreme Court seats Republican senator says plans to confirm justice before election 'completely consistent with the precedent' MORE (D-N.Y.) expressing concerns about the measure in the hope that the two senators, who spearheaded it, would be willing to amend the law in the future.

"We would hope to work with you in a constructive manner to appropriately mitigate those unintended consequences," the senators wrote.

The White House on Thursday accused lawmakers of experiencing buyers remorse.

White House spokesman Josh Earnest called it “an abject embarrassment” that members are considering potential changes to the measure so soon after its passage.

“I think what we’ve seen in the United States Congress is a case of rapid-onset buyer's remorse," he said.

"There's a desire to amend what occurred yesterday to put us in a better place,” said Sen. Bob Corker Robert (Bob) Phillips CorkerHas Congress captured Russia policy? Tennessee primary battle turns nasty for Republicans Cheney clashes with Trump MORE (R-Tenn.), the chairman of the Senate Foreign Relations Committee.

Schumer told reporters Thursday that he was open to looking at potential changes but “not any that hurt the families."

Cornyn said Thursday that he and Schumer worked with the State Department and lawmakers, including Corker and Graham, to make changes to the legislation that addressed concerns before the Senate initially passed the bill in May.

A Cornyn aide — noting that lawmakers dropped their holds on the bill after the changes — said a provision was added that allows the U.S. government to pause a lawsuit against another country if the State Department verifies the administration "is engaged in good faith discussions" about resolving the dispute. Another change made at the request of Graham would strengthen requirements that a potential defendant must be able to prove that a foreign government's actions were "more than negligent, i.e. reckless or intentional."

Scott Wong and Mike Lillis contributed.