The potential changes could make it more difficult for the families to pursue lawsuits but could also make it harder for the US to be sued for alleged wrongdoing.
"I have tremendous empathy for the victims," said Sen. Bob Corker of Tennessee, the chairman of the Foreign Relations Committee. "At the same time, I have concerns about the precedent this bill will set and what it may mean to American servicemen and women and others.
"Let's face it: our alleged drone attacks have killed civilians in Pakistan," Corker said. "Our alleged drone attacks have killed civilians in Afghanistan. And I think once you start opening the door for these types of activities, it can be very problematic."
The late push from the two senators -- who in May held up a final vote on the bill but eventually allowed it to pass on a voice vote -- comes as President Barack Obama is expected to veto the legislation in the coming days. The House passed it just last week.
Corker said he hopes a congressional attempt to override a veto won't come until after the election so there is time to negotiate changes that would be satisfactory to both sides.
Sen. Lindsey Graham of South Carolina said he is not convinced the Saudi government is culpable even though many of the 9/11 hijackers were Saudi citizens.
"They want to blame bin Laden," he said about the Saudis. "All I can say is, from what I can tell, I don't think the government of Saudi Arabia was involved here."
Graham, who said he's in close contact with Saudi officials about the issue, warned it could destroy America's relationship with Saudi Arabia, a critical ally in the tumultuous Middle East.
"This is an odd situation in the sense that 9/11 families are high on everybody's list to take care of," Graham said. "It comes at a time when Saudi Arabia believes that America is not a reliable ally. It comes at a time when they think they're being blamed for things they didn't do. All I'm trying to find is a way to move forward with a legal process that doesn't destroy the relationship. That's worth investing some time in."
The bill, known as Justice Against Sponsors of Terrorism Act, JASTA, would prevent countries accused of terrorism on American soil from claiming sovereign immunity against lawsuits.
Graham acknowledged it might be too late to push through changes.
"I don't know if we can get there or not. But I do know this, that Saudi Arabia views this as one of many problems and this could be the breaking point. I don't think they're bluffing. Time will tell," he said.
A group of 9/11 family members lobbying for the bill blasted Corker and Graham in no uncertain terms.
"After a visit to the halls of Congress by an army of Saudi lobbyists led by the Saudi Foreign Minister, JASTA supporters Senator Bob Corker and Senator Lindsey Graham called today for the JASTA veto override vote to be delayed for two months," the families said. "All it took -- apparently -- was a visit by the Saudi foreign minister to get these senators to forsake the families and seek a two-month delay in a vote to override the president's imminent veto."
Congressional supporters of the bill said they are confident Congress will override the President's veto.
"It passed by unanimous consent in the Senate and it passed unanimously in the House," noted Sen. John Cornyn of Texas, the Republican whip who co-authored the bill with Sen. Chuck Schumer of New York, who is in line to be the next Democratic leader.
Sen. Dick Durbin of Illinois, the Democratic whip, said he would consider proposals from Graham and Corker, but in the meantime, the White House would have a difficult time getting the 34 vote it needs in the Senate to preserve the President's veto.
"For the President to come up with 34 who will stand by him on this, he's got a lot of work ahead of him. Most people believe Saudi Arabia and others should be held accountable if they've been guilty of any terrorists activities. So I'm open to any changes or fixes but I think there is pretty strong bipartisan sentiment in support," Durbin said.