Sen. Jeff Flake, R-Ariz., told reporters at a breakfast sponsored by The Christian Science Monitor the president has been helpful so far, The Hill said.
"I think he's played the right role so far -- the outside cheerleader, if you want to call it that -- that's been I think useful and effective," Flake said. "As a Republican, it would be more difficult were he out front and pushing this issue."
Sen. Michael Bennet, D-Colo., agreed: "I think that he's been constructive in this process, allowing us to see how far we can get on the legislation, negotiating it together. I expect us to continue to do that on the floor. If things could continue the way they've been, that would be very positive."
The Senate has voted to begin debate on the bill. Its backers say they believe they have the 60 votes needed to get it to the floor but want a bigger margin, at least 70 votes, to increase support among House Republicans.
Notable deaths of 2014 [PHOTOS]