On CNN, Conway said Trump has "great faith" in the USA intelligence community.
Republicans have "sought to expose our client's identity which could jeopardise their safety, as well as that of their family", Zaid wrote on Twitter.
Trump's comments on Sunday came on the heels of the whistleblower's attorney, Mark Zaid, saying his client would answer questions directly to Republicans on the intelligence committee leading the impeachment inquiry.
Zaid said the whistleblower would answer questions directly from Republican members "in writing, under oath & penalty of perjury".
Lawyer Mark Zaid tweeted that he would not entertain questions that "seek identifying info" about the man, a Central Intelligence Agency officer who was formerly tasked to White House national security functions.
Later outside the White House, Trump called the whistleblower an "Obama guy" and a fraud and said, "The whistleblower should be revealed". "That is not our role".
News of the whistleblower's offer to provide written testimony came three days after the House voted to formalize the impeachment inquiry into President Trump and set in motion a timetable for public hearings. Besides, the White House allegedly tried to cover it up. That call became the catalyst for the impeachment inquiry.
"I always got along with him", Trump said Sunday, misrepresenting his relationship with the renowned Washington bruiser in a possible sign of concern about his potential testimony.
Pollsters also found that Trump's job approval stands at 45 percent, while 53 percent say they do not approve of his performance in the White House. "You just don't want to report it", Trump told reporters at the White House. On the other hand, Trump said the whistleblower needs to testify.
"This is over a phone call that is a good call. It was totally appropriate". Oppositely, Minority Leader Kevin McCarthy of California quoted Alexander Hamilton's warning of impeachment as a "partisan tool" in the Federalist Papers.
In October, Republicans on committees main the impeachment inquiry sent a letter to Dwelling Intelligence Committee Chairman Adam Schiff disturbing he bring the whistleblower in to testify publicly.
Sen. Marco Rubio (R-Fla.) added that he wasn't sure if the panel was going to talk to the whistleblower, and he questioned the value of receiving any testimony from the individual.
The House last week passed a resolution to formalise the impeachment inquiry.
For only the fourth time in US history, the House of Representatives has started a presidential impeachment inquiry. "Let the Senate Intel Committee - they were given this task not me, you know those two guys have worked pretty well together - let the Senate do its thing", Graham said.
Persistently slow wage growth appeared to be a main driver of discontent, with 36 per cent of those who said they were worse off blaming their income levels.
House Majority Leader Steny Hoyer, who also appeared on CBS on Sunday, said Congress will move "relatively soon" to hold public hearings but would not commit to wrapping up those hearings before Thanksgiving, which falls on November 28 this year. On Friday, the United States labour department said average hourly income had risen 3 per cent in October, growth that was near highs for the past decade but lower than before the financial crisis.