Sources say the pressure is directed at members who might support a bill proposed by Sens. Lindsey Graham, R-South Carolina, and Dick Durbin, D-Illinois, and rest of the so-called Gang of Six.
Two GOP sources with knowledge of the process say Republican leadership -- including Sen. Mitch McConnell -- has made clear to several members that it wouldn't be helpful to the current Republican strategy to get onto the proposal.
Meanwhile, Congress remained hours away from a government shutdown on Friday.
Republican Majority Whip John Cornyn's spokesman denied any active whip operation, pointing to Cornyn's public statements against the bill.
"To write our office is whipping against their proposal (I haven't seen a bill, have you?) is factually inaccurate. It could not be further from the truth," the spokesman said in an email.
But sources close to the process say the pressure does exist.
Durbin and Graham have managed to lock up four Republicans -- Sens. Lisa Murkowski of Alaska, Lamar Alexander of Tennessee, Mike Rounds of South Dakota and Susan Collins of Maine -- in addition to the three Republicans that made up half of the "gang" to support the proposal so far -- still four votes short of enough to muster the 60-vote threshold needed to advance the legislation if all Democrats join in. It still has not been introduced in legislative text form, but a description of the pieces of the bill has been made public.
President Donald Trump last week rejected the proposal, in vulgar terms
, in an Oval Office meeting where it was presented to him.
The day before the Gang of Six unveiled their plan, Graham, Flake and Durbin were visibly working the Senate floor during a long FISA cloture vote. The three of them spoke at separate times with Sen. Bob Corker of Tennessee, for example, who voted for immigration reform before.
Asked if anyone was trying to undermine their efforts on the bill on Thursday, Republican Sen. Cory Gardner of Colorado, himself a member of the GOP leadership, denied activity to tank a deal.
"I think there are probably people on the left and right who don't want a solution; I don't feel that anybody is actively working against a solution," Gardner told reporters.
Graham and Cornyn have landed publicly on different sides of the issue. Asked about Cornyn's assertion there would not be a short-term continuing resolution of a few days on Thursday, Graham replied: "Well then you need to tell Sen. Cornyn, I respect him a lot, but he needs to get 60 votes. Good luck."
Government funding runs out at midnight on Friday and the Senate does not appear to have the votes for a House-passed continuing resolution into mid-February.