Congress in November could weigh changes to a law allowing families of victims of terrorism on U.S. soil to sue foreign governments.

On Wednesday, lawmakers overwhelmingly overrode President Barack Obama’s veto of the Justice Against Sponsors of Terrorism Act, which would allow families of 9/11 victims to sue Saudi Arabia over alleged ties to the attacks. But lawmakers, wary of unintended legal consequences, are pondering changes to the statute.

“I think it was an example of an issue we should have, on a bipartisan basis, talked about much earlier,” Senate Majority Leader Mitch McConnell of Kentucky said at a Thursday press conference. “Everybody was aware of who the potential beneficiaries were, but nobody really focused on the potential downsides in terms of our international relationships.”

Opponents of the legislation argue that the law could make the U.S. vulnerable to reciprocal actions, since it alters the concept of sovereign immunity, which holds that a state cannot be subject to a lawsuit.

Minority Leader Harry Reid was the lone senator to vote to sustain the president’s veto. His office pointed to a letter Obama had sent to the Nevada Democrat when asked for an explanation on his vote. Reid declined to elaborate on his vote during a Thursday press conference.

[Override of 9/11 Bill Veto Put Some Republicans in a Quandary]

South Carolina GOP Sen. Lindsey Graham told reporters Wednesday that there are 20 senators discussing a “fix” to address the issue of sovereign immunity.

Senate Foreign Relations Chairman Bob Corker of Tennessee said the number is higher. But both he and Graham acknowledged that reaching a final agreement to tweak the new law and pass it during the lame-duck session after the elections would be an uphill climb.

Asked Thursday if the Senate would consider any changes during the lame-duck session, McConnell said changes were “worth further discussion.”

House Minority Whip Steny H. Hoyer declined to say whether he would be open to allowing changes to the law. 

“There are valid concerns and just because we passed the bill, those concerns should not be ignored, and we need to pursue those,” the Maryland Democrat said Thursday.

White House Press Secretary Josh Earnest termed the Senate’s 97-1 override vote “the single most embarrassing thing that the United States Senate has done” in more than 30 years. At a CNN town hall Wednesday night, Obama called the override a “political vote.”

But lawmakers chided the White House for not engaging with lawmakers about their opposition to the bill.

“I think it was just a ball dropped,” McConnell said. “I wish the president — I hate to blame everything on him and I don’t — but it would have been helpful had he, we, had a discussion about this much earlier than last week.”

[Obama Vetoes Terrorism Bill as Override Votes Loom]

Corker took to the floor Thursday afternoon to say that he had attempted to set up a meeting between the president, Senate leaders, and the primary sponsors of the law, Democratic Sen. Charles E. Schumer of New York and Senate Majority Whip John Cornyn of Texas. Corker said the White House did not seem interested.

“For reasons that still are unknown to me, that [meeting] was not achievable,” he said. “There was no desire whatsoever to sit down and meet. I’m unaware of any meetings that took place to try and resolve this issue.”

But Corker said he was hopeful, now that the bill is law, that the major players would be willing to sit down to “create some alterations.”

Schumer said Thursday that he was open to listening to any proposal, but said, “It has to be something that doesn’t weaken the bill and limit the right for these families to get their day in court and justice.”

Schumer pointedly denied that the override vote was political, as the president had suggested.

“I look at the families, it’s hardly political for me. I’ve sat and worked with these families for five years. I feel their pain,” Schumer said. “Not close to the amount because I didn’t lose a loved one the way they did, but this is about justice.”

John T. Bennett and Lindsey McPherson contributed to this report.

Contact Bowman at bridgetbowman@rollcall.com and follow her on Twitter @bridgetbhc.