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.) opened the door Thursday to changing legislation that allows families of 9/11 victims to sue Saudi Arabia in U.S. court.

"I do think it's worth further discussions, but it was certainly not something that was going to be fixed this week," he told reporters when asked about a push by some senators to tweak the measure.

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Across the Capitol, 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.) 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 had 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), was turned into law on Wednesday after Congress overrode President Obama's veto. It was the first veto override of Obama's presidency.

Despite the override, 28 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.

Sen. Lindsey Graham Lindsey Olin GrahamGraham: GOP will confirm Trump's Supreme Court nominee before the election Fox's Napolitano: Supreme Court confirmation hearings will be 'World War III of political battles' Grassley, Ernst pledge to 'evaluate' Trump's Supreme Court nominee MORE (R-S.C.) said Wednesday that he would like the Senate to take up changes to JASTA as early as the lame-duck session. The Senate is leaving town Thursday until after the November elections.

"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.

Though the Senate voted on Wednesday to overwhelmingly nix the president's veto in a 97-1 vote, some lawmakers said they had misgivings about the bill. They stressed, however, that any push to find an alternative was largely ignored by the Obama administration.

McConnell echoed that sentiment Thursday, calling the legislation "an example of an issue that we should have talked about much earlier."

"You know, that was a good example of — it seems to be a failure to communicate early about the potential consequences of a piece of legislation was obviously very popular," he said.

- Scott Wong and Mike Lillis contributed.