The goal of the senators is to reach bipartisan consensus on a deal that could be on the floor as soon as Monday, according to members involved in the effort.
"There's a consensus among a lot of us that we want to suspend arms sales and stop support for the war -- how do you best do that?" said Sen. Lindsey Graham, a South Carolina Republican who is directly involved in the effort.
Asked if he's spoken to the White House about the push, Graham said bluntly: "They know where I'm at."
"It'll have teeth," said Senate Foreign Relations Chairman Bob Corker, a Tennessee Republican, who added that the senators are trying to piece together various proposals into one plan to achieve bipartisan consensus. "We're trying to get everything together so we do something constructive."
Corker also said CIA director Gina Haspel gave the "most precise presentation I've ever heard in 12 years" in the Senate when she briefed senators Tuesday on the Khashoggi murder.
He said a colleague corrected his comments Tuesday that a jury would convict bin Salman in 30 minutes. It would have been "20 minutes," he said the colleague had told him.
Corker said he would be "benevolent" and give Defense Secretary James Mattis and Secretary of State Mike Pompeo the benefit of the doubt for their prior briefing, saying they may be downplaying a link with the crown prince because they haven't seen the same intelligence that the CIA director has.
On Wednesday, Graham -- along with Sens. Marco Rubio, R-Florida, Dianne Feinstein, D-California, Ed Markey, D-Massachusetts, Todd Young, R-Indiana, and Chris Coons, D-Delaware -- introduced a strongly worded resolution condemning the crown prince for a range of actions, including the crisis in Yemen, the blockade in Qatar, the jailing of dissidents and the killing of Khashoggi. The measure, while nonbinding, is a symbolic rebuke -- and it could be added to a broader package targeting Saudi Arabia.
"Those who suggest we must sacrifice our principles for security will have neither," said Young.
Without a bipartisan deal, Corker warned, a measure drafted by Sens. Bernie Sanders, a Vermont independent, Mike Lee, a Utah Republican, and Chris Murphy, a Connecticut Democrat, to end US support within 30 days for the war in Yemen would pass the Senate as soon as next week.
"If there's not (a deal), the other one looks to me like it's going to get 51 votes," he said.
Murphy said Wednesday that he supports doing something in the measure to punish Saudi Arabia over the Khashoggi murder but his primary concern is stopping the horrific humanitarian crisis in Yemen.
"The most important thing is to stop the slaughter of civilians in Yemen," Murphy said. "I'm upset about Khashoggi but I'm much more upset about that Congress has allowed thousands of kids to die inside Yemen. We need to end that war and if the resolution doesn't end US participation in the war in Yemen then there are not 50 votes for the final product."
He said he will meet with Corker and others Thursday to discuss the scope of the legislation.
But with only limited time left in the lame-duck session, GOP leaders are wary about bringing up the measure because it could dominate the floor schedule next week. Because of the unusual process, bringing up the Sanders-Lee-Murphy measure -- without a bipartisan deal to consider amendments -- could open up the floor to a session known as a "vote-a-rama" that could open up the bill to a wide range of unrelated issues.
"The conundrum is if we get on it without an agreement on amendments then we've got a vote-a-rama," said Senate Majority Whip John Cornyn, a Texas Republican. "And so there's no germaneness requirement, so everything from criminal justice reform to repeal-and-replace Obamacare to 'Medicare-for-all' are all fair game. That's the conundrum."
Cornyn added: "I think we should condemn the murder of Khashoggi in the most emphatic terms but I don't think we should cut our nose off to spite our face with regard to the proxy war in Yemen against Iran."
This story has been updated to include additional developments.