“The pushback on that is coming from the Republicans,” said Senator Harry Reid of Nevada, the minority leader.

But support from other Republicans and many Democrats seems to be giving the bill momentum.

“Certainly they have been a partner in many ways and that has been longstanding,” said Senator Bob Corker, Republican of Tennessee and the chairman of the Senate Foreign Relations Committee. “On the other hand we understand that the whole Wahhabi effort emanated from there and that alone is an issue,” he said, referring to the radical strain of Islam practiced in the kingdom.

But prospects for the bill are more uncertain in the House. In a news conference on Tuesday, Speaker Paul D. Ryan, who recently visited Saudi Arabia, said he would review the bill but needed to make sure “we’re not making mistakes with our allies.”

The Saudi government, which has long denied any involvement in the Sept. 11 attacks, has warned that it might liquidate hundreds of billions of dollars worth of American assets if the bill becomes law. Adel al-Jubeir, the Saudi foreign minister, told lawmakers and Obama administration officials during a visit to Washington last month that such a move could be necessary to avoid the assets being frozen in court cases brought by families of Sept. 11 victims.

Those cases, some of which have tried to hold members of the Saudi royal family and Saudi charities liable for what the plaintiffs allege was financial support for terrorism, have been largely stymied because of a 1976 law that gives foreign nations broad immunity from American lawsuits. The current legislation would amend the law, allowing for nations to be sued in American courts if they are found to have played any role in terrorist attacks that killed Americans on home soil.

There is little question that relations with Saudi Arabia have frayed in recent years, in part because of a perception in Riyadh that the Obama administration has abandoned its traditional allies in the Middle East. But some experts said that the irony of the growing anti-Saudi sentiment in Congress is that the Saudi government in recent years has scaled back its support for extremist groups in the Middle East.