Senate passage of the Yemen resolution could prompt Mr. Trump to issue the first veto of his presidency, and it would come after Republicans have registered their unhappiness over other foreign policy issues, such as the president’s plan to withdraw troops from Syria and Afghanistan and his threats to pull the United States from the North Atlantic Treaty Organization. On Wednesday, a bipartisan group of senators introduced new sanctions on Moscow that would require the secretary of state to submit a determination of “whether the Russian Federation meets the criteria for designation as a state sponsor of terrorism.”

Democrats demanded Senate action.

“This is their opportunity to send a message to the Saudis that their behavior on Khashoggi and their flagrant disregard of human rights is not consistent with the American way of doing business and not in line with American values,” Representative Ro Khanna, Democrat of California, the bill’s lead sponsor, said in an interview, adding that he was “relieved” that Congress finally took action on the resolution, which he first introduced in 2017.

The House resolution is a rare use of the 1973 War Powers Act, which gave Congress the ability to compel the removal of military forces absent a formal declaration of war. Those powers, created in the wake of the Vietnam War, have almost never been used, as lawmakers have demurred from intervening in politically sensitive matters of war, peace and support for the troops.

But the conflict in Yemen is proving to be different. Senators sponsoring their own resolution are expected to act quickly to force a vote, as lawmakers in both parties fume over how the administration has responded to Saudi Arabia’s role in the killing of Mr. Khashoggi, who was based in Virginia. Senator Christopher S. Murphy, Democrat of Connecticut and one of the sponsors, said he anticipated a vote “within the next 30 days.”

The White House pre-emptively threatened to block the resolution over the weekend, with administration officials arguing in a statement of administration policy that “the premise of the joint resolution is flawed” because the United States has provided only “limited support to member countries of the Saudi-led coalition” in Yemen.