WASHINGTON (Reuters) - U.S. lawmakers expressed doubts on Thursday 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 the Democratic president for not consulting them adequately. “I do think it 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. troops in particular. Ryan did not give a time frame, but Republican Senator Bob Corker, chairman of the Senate Foreign Relations Committee, said he thought JASTA 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 seeking damages from the Saudi government. Riyadh denies longstanding suspicions that it backed the hijackers who attacked the United States in 2001. Sept. 11 families lobbied intensely for the bill, getting it passed by the House days before the 15th anniversary of the 2001 attacks earlier this month after years of effort. “We have to understand the political environment we’re in right now and the tremendous support the 9/11 victims have in the United States,” said Robert Jordan, a former U.S. ambassador to Saudi Arabia. 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. The Saudis lobbied furiously against JASTA, and the Saudi foreign ministry condemned its passage in a statement on Thursday. “The erosion of sovereign immunity will have a negative impact on all nations, including the United States,” said the statement, which was carried on state news agency SPA. Still, the new law is not expected to have a lasting effect on the two countries’ strategic relationship. Saudi-U.S. ties have endured “multiple times of deep outrage” over 70 years, said Thomas Lippman of the Middle East Institute. “The two countries need each other as much today as they did before the day before yesterday,” he said. White House spokesman Josh Earnest mocked lawmakers for shifting “within minutes” from overwhelmingly voting to override Obama’s 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 said he had tried to work out a compromise with the White House, but Obama administration officials 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.  He said he would oppose a suggestion that the measure be narrowed to only apply to the 2001 attacks on Washington and New York. “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. He said the Saudis were been willing to work on a compromise, and denied they had threatened retaliation. 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. “I do feel passionately this is a mistake for a variety of reasons, in terms of threats to troops, diplomats, sovereignty, there’s serious problems here. Hopefully we can find a way to change the tenor of this,” Lott said. 