"There are many times he has expressed, I'm told by people in the White House, the desire to fire Mueller, the desire to pardon people under investigation including his family," Bernstein, a CNN contributor, told CNN's Dana Bash on "State of the Union." "His lawyers are telling him what he wants to hear -- that's what I'm told -- by lawyers in the White House, they're telling him what he wants to hear to keep him from acting precipitously and to go off and fire Mueller in a rage, or fire (Deputy Attorney General) Rod Rosenstein in a rage. They have an out-of-control client."
Bernstein added: "The President of the United States, in their view, is out of control most of the time, in their view, when it comes to this investigation."
Bernstein's partner of Watergate reporting fame, The Washington Posts' Bob Woodward, appeared alongside him on the program, telling Bash he expects Mueller to leave no stone unturned in the investigation.
"Mueller is going to do this job," Woodward said. "I think some of the Trump lawyers are fantasizing when they think it's going to be over quickly. There is no reason the Mueller operation is going to close up; they're going to want to be thorough."
When asked earlier on the show about the Russia investigation, former White House communications director Anthony Scaramucci said Trump isn't worried.
"I really don't think he's worried about it," Scaramucci told Bash. "When you really uncover the data and the information, you'll find that the President was nowhere near the Russians as it related to the campaign, the election or anything thereof."
Woodward and Bernstein also said they didn't think California Democratic Sen. Diane Feinstein's remarks on NBC's "Meet the Press" earlier this month, in which she said she saw the outlines of an obstruction of justice case forming against Trump, were accurate.
"I'm not sure Sen. Feinstein is quite right," Woodward said. "It's possible this will lead to an obstruction of justice case, but we have not seen the evidence."
Trump versus Nixon
Woodward and Bernstein drew parallels between the Trump administration and Nixon administration, which they famously covered during their time together at The Washington Post.
They both said Trump's behavior toward the press is similar to Nixon's during his presidency.
"Nixon in Watergate tried to make the conduct of the press the issue instead of the conduct of the President and the men around him," Bernstein said. "Donald Trump has gone even farther; he's tried to undermine the credibility of the press as a national institution, to the detriment of the country, by these broad attacks on the press."
Woodward defended the press but said that some political commentators haven't been fair in their remarks about the President.
"The tone is a big issue here," he said. "And lots of reporting -- particularly in television commentary -- there's a self righteousness and smugness in people kind of ridiculing the President."