WASHINGTON (Reuters) - U.S. lawmakers on Thursday expressed doubts about Sept. 11 legislation they forced on President Barack Obama, saying the new law allowing lawsuits against Saudi Arabia could be narrowed to ease concerns about its effect on Americans abroad.

A day after a rare overwhelming rejection of a presidential veto, the first during Obama’s eight years in the White House, the Republican leaders of the Senate and House of Representatives opened the door to fixing the law as they blamed Obama, a Democrat, for not consulting them adequately.

“I do think is worth further discussing,” Senate Majority Leader Mitch McConnell told reporters, acknowledging that there could be “potential consequences” of the “Justice Against Sponsors of Terrorism Act,” known as JASTA.

House Speaker Paul Ryan said Congress might have to “fix” the legislation to protect U.S. service members in particular.

Ryan did not give a time frame for addressing the issue, but Republican Senator Bob Corker, chairman of the Senate Foreign Relations Committee, said he thought the issues could be addressed in Congress’ “lame-duck” session after the Nov. 8 election.

The law grants an exception to the legal principle of sovereign immunity in cases of terrorism on U.S. soil, clearing the way for lawsuits by the families of victims of the attacks seeking damages from the Saudi government. Riyadh has denied longstanding suspicions that it backed the hijackers who attacked the United States in 2001. Fifteen of the 19 hijackers were Saudinationals.

Riyadh is one of Washington’s longest-standing and most important allies in the Middle East and part of a U.S.-led coalition fighting Islamic State militants in Iraq and Syria.

JASTA will add tension to U.S.-Saudi relations, after friction over Obama’s 2015 nuclear deal withSaudi rival Iran.

White House spokesman Josh Earnest noted how quickly lawmakers shifted from overwhelmingly voting to override the veto to wanting to change the law.

“I think what we’ve seen in the United States Congress is a pretty classic case of rapid onset buyer’s remorse,” Earnest told a White House briefing.

Corker criticized the White House, saying he had tried to work with the administration to find a compromise before the veto override, but the administration declined a meeting.

Democratic Senator Chuck Schumer, who championed JASTA in the Senate, said he was open to revisiting the legislation.

“I’m willing to look at any proposal they make but not any that hurt the families,” he said at a news conference.

However, he said he would oppose a suggestion that the measure be narrowed to only apply to the Sept. 11, 2001, attacks.

“You know what that does? It tells the Saudis to go ahead and do it again, and we won’t punish you,” Schumer said.

Corker said another suggestion was establishing an international tribunal so experts could determine whether there was culpability, and that the Saudis had been willing to sit down and work on a compromise.

“Never in their conversations has there been any kind of threat,” he said. “They are making observations about where this could lead. We’ve had other Arabs in the region weigh in and express concerns.”

Trent Lott, a former Republican Senate Majority Leader now at a Washington law firm lobbying for the Saudis, said attorneys would look carefully at JASTA’s language.

“You can amend something like this,” Lott said.

(Additional reporting by Roberta Rampton, Susan Cornwell, David Morgan, Yara Bayoumy, David Alexander and Susan Heavey; editing by Grant McCool)