Republicans have worried publicly and privately that they are losing the war of appearances in the battle over the cliff.
On Thursday, another poll showed Republicans may have reason to worry about public perception. A Quinnipiac University survey found respondents trust Obama and Democrats more than Republicans on the cliff talks by a wide margin - 53 percent to 36 percent.
In both public statements and private encounters, Obama has tried to encourage Republicans wavering from the position of the party leadership.
Republican Representative Tom Cole, who last week broke ranks with his party and agreed to accept higher tax rates on the richest Americans, said Obama took him aside at a White House Christmas party on Monday and joked about the criticism Cole had received from Republicans.
"The president pulled me over and he said, 'Cole, come closer, I want to see the bruises,'" Cole told Reuters. "He said, 'Seriously, I will go further on this thing than you guys think. I know we can get something done.'"
While other Republicans have questioned Obama's commitment, Cole said, "I take him at his word," adding: "The best is to get to that discussion as quickly as we can."
Obama, meanwhile, played to his strengths with the latest in a series of the sort of public events he has used against Republicans in the fiscal cliff fight: a visit with a family in the Virginia suburbs of Washington to illustrate how Republican tax proposals would hurt the middle class.