Why it matters: Bipartisanship is in short supply in the Senate. Graham's sudden death may briefly revive it.
Senate Majority Leader John Thune (R-S.D.) said passing the updated sanctions bill — which has 85 cosponsors — "would be a great tribute to the legacy of Lindsey."
"I urge Senator Thune, in honor of Lindsey, to put the Russia sanctions bill on the floor immediately," Senate Minority Leader Chuck Schumer (D-N.Y.) said. "It will pass overwhelmingly and help our allies in Ukraine."
"This would be a good way to show us in a bipartisan way coming together on something that he was literally using some of his last breaths to fight for," Sen. Cory Booker (D-N.J) said.
Driving the news: The mechanics of replacing Graham are moving quickly.
But then Trump would urge Republicans to hold off, giving his administration room to pursue its own pressure campaign against Russian President Vladimir Putin.
In a meeting in Ankara, Turkey, late last week with Treasury Secretary Scott Bessent, Graham and other senators emerged convinced they had resolved the administration's concerns. Graham briefed Trump on their progress in a Saturday night phone call.
"We are proud to announce that we have reached an agreement with the Trump Administration to move our updated Russia sanctions legislation forward," Graham, Blumenthal and Sens. Jeanne Shaheen (D-N.H.) and Roger Wicker (R-Miss.) said in a joint statement.
Zoom in: The spirit of the updated bill remains the same, but the substance differs.
The initial bill would have imposed tariffs of up to 500% on countries, including China, that continue purchasing Russian oil and gas. The updated version narrows the scope of those penalties.
Between the lines: Senators continued to grapple Monday with Graham's unexpected death.
Thune fought back tears during remarks on the Senate floor as colleagues reflected on Graham's influence.
"There are no words to describe his impact on the foreign and domestic policy of the United States," Wicker said.
Blumenthal recalled his final extended conversation with Graham over the weekend.
"He exulted at reaching an agreement on our Russian sanctions bill and said, 'This is a big effing deal.'"
The bottom line: The original sanctions bill has already cleared the House, but they also want to change it.
"It would be very appropriate to name this the 'Lindsey Graham Sanctions Act,'" said Rep. Joe Wilson (R-S.C.). "The way to restrain Putin is indeed to bankrupt the oligarchs."