“The families have been asking for this for over a decade,” said Terry Strada, whose husband, Thomas S. Strada, was killed in the attack on the World Trade Center towers. She has long lobbied Congress on the issue. “We don’t feel this is fast-track in any way, shape or form,” she said.

Mr. Obama has voiced opposition to the effort for months, fearing that it could expose the United States to lawsuits by people in other countries. A White House official said this week that the administration’s position has not changed even after some changes were made to the bill to mollify critics.

The bill “is a politically cost-free way for Congress to send a signal of seeming seriousness about terrorism on the dawn of the 15th anniversary of 9/11,” said Jack Goldsmith, a professor of law at Harvard who served in the Department of Justice under President George W. Bush. “Congress itself could have investigated lingering questions about 9/11, but instead is delegating those tasks to the unelected judiciary. The costs of the law will be borne by courts, which are an awkward place to ascertain Saudi responsibility for 9/11, and especially the president, who will have to deal with the diplomatic fallout with Saudi Arabia and other nations.”

The bill addresses a 1976 law that gives other countries broad immunity from American lawsuits. It amends the law to allow for nations to be sued in federal courts if they are found to have played any role in terrorist attacks that killed Americans on home soil. It also allows Americans to direct financial claims against those who funded the attacks.

The administration has argued that it would put Americans at legal risk overseas. That position seemed at least somewhat validated when Pierre Lellouche, a member of the French Parliament who is chairman of the rough equivalent in France of the Senate Foreign Relations Committee, said he would pursue legislation that would permit French citizens to sue the United States with cause.