The source said Comey was concerned that the allegation would make the FBI look bad, and that concern was part of what prompted the FBI director to have his staff reach out to staff at the Justice Department asking them to knock down the allegation.
The source said Comey felt "institutionally he has to push back on this" because the magnitude of the allegations that Comey knows not to be true.
Over the weekend, senior-level officials within the FBI reached out to career personnel at the Justice Department to try to clear up reports that Obama wire-tapped Trump's phone, a US official told CNN. Part of the discussion was the FBI asking for DOJ to publicly knock down the allegations, which so far DOJ has not done.
Comey was aware of those discussions, the US official said, but Comey himself did not reach out to DOJ over the weekend.
Comey was frustrated that his request for DOJ to knock down the story went unanswered over the weekend, according to the source familiar with the matter, and the FBI director is still trying to figure an appropriate channel for the FBI and DOJ to formally come out and say the allegations are not true.
"He might come out and say only time will tell, but right now they (FBI and DOJ) should be in the exact same place and they're not," the source said.
The source added that Comey as of now has no plans to resign.
"Does he know of possibility there might be a confrontation and be fired by the President? Sure," the source said. "Does he worry about it? No."
The Justice Department and FBI had no comment.
On CNN's "Situation Room," Homeland Security Secretary John Kelly said Trump must have evidence of his wiretap claims
, and Comey, who he calls a "friend," should turn the allegations over to "an investigative arm" to get to the truth of what happened.
"Jim Comey is an honorable guy," Kelly said. "And so is the President of the United States. And the President must have his reasons."
White House press secretary Sean Spicer said Monday that he's "almost 100% certain" that Trump and Comey haven't spoken since the President made the wiretap allegations, but stopped short of saying definitively that Trump retains confidence in his FBI director.
"There's nothing that I have been told by him that would lead me to to believe that anything is different than what it was prior," Spicer said.
Later, asked twice in an interview with CNN's Sara Murray about Trump's feelings toward Comey, Spicer again demurred.
"I haven't asked him that yet. Obviously he's focused today first and foremost on this effort to keep the country safe," he said.
Spicer said he would let Comey speak for himself when it comes to his potential concerns about Trump's allegations of wiretapping.
Sen. Lindsey Graham, R-South Carolina, told reporters Monday evening that the wiretapping "would be earth-shattering if it happened," adding that he did not know of any evidence to support the President's claim, but agreed with Trump that Congress should investigate the matter.
Sen. John McCain called on Trump to provide evidence for his assertion.
"I'm all in favor of Congress continuing investigation," the Arizona Republican said. "But first I believe that the President should tell the American people what evidence he has that this kind of action was carried out by the previous president."
Rep. Chris Stewart, a member of the House Intelligence Committee, made a similar point to CNN's Erin Burnett on "OutFront." The Utah Republican said it was a very serious accusation, worthy of congressional inquiry, but he also called on Trump to use his authority to declassify any existing information that could bolster his currently unsubstantiated case.
"I think the quicker this type of information comes out, the better it is," Stewart said. "If the President has information and he could declassify that without endangering national security, I would encourage him to do that."