Sen. Cory Gardner, R-Colo., said he believed people on both sides of the debate had little interest in actually resolving the current stalemate on immigration policy, specifically the Deferred Action on Childhood Arrivals policy.

Gardner nevertheless said he believed support for a solution was broad enough for a deal to be possible.

"I have no doubt that there are probably people on the Republican side of the aisle who may not want a solution on this. There are probably some people on the Democratic side who don't want a solution on this. They'd rather have those things to fight about, to fight over. But that's not where the Democrats I have been working with are. That's not where the Republicans I have been working with are," Gardner told CBS News on Sunday morning.

Gardner said he had had numerous conversations with Trump in which "he's talked about the need to bring forward a solution... I do think he is sincere."

The senator added he "had no reason to doubt" that President Trump wanted one as well "despite the blowout that happened this week," referring to reports that Trump referred to Haiti and African nations during a closed-door meeting with senators as "shithole" countries. Trump and some Republicans present have denied the reports.

Sens. Tom Cotton, R-Ark., and David Perdue, R-Ga., both of whom were present at the meeting where Trump reportedly made the remarks, said Sunday that they did not hear Trump say them. Both made a point of saying that Sen. Dick Durbin, D-Ill., who was the first to affirm reports that Trump made the comments, had a history of publicly twisting comments made by Republicans in private meetings.

Sen. Lindsey Graham, R-S.C., who was also at the meeting, has backed up Durbin's account. Sen. Jeff Flake, R-Ariz., was not present but told ABC News Sunday that senators who were present gave him an account similar to Durbin's and Graham's.