Veteran journalist Carl Bernstein said Sunday the Republican Party's future could depend on whether it starts pushing back against falsehoods espoused by President Trump.

"Republicans, if they want to remain a viable party in the United States, are going to have to say, 'Look, the facts are adding up in a really ugly way,' which they are so far," Bernstein, who broke the Watergate scandal with former Washington Post colleague Bob Woodward, told CNN during an interview on "Reliable Sources."

[Also read: Trump: NBC News 'now as bad as Fake News CNN' after false Michael Cohen bombshell]

Bernstein made the comments in the context of drawing comparisons between the embattled Nixon administration and the Trump White House.

"There's huge differences between Watergate and what we're seeing here, but there is one similarity and that is trying to make the conduct of the press the issue," Bernstein said.

"What the press did in Watergate is it kept covering the story, the cover-up unraveled, the facts became known and partly because Republicans, who were in and willing to engage in a bipartisan investigation as they are not now, were swayed by the facts," he continued.

But Bernstein said the Washington press corps should moderate its coverage of Trump's misstatements.

"I don't think we need to engage every moment in saying, 'Liar, liar, liar, your pants are on fire. What we need to do is keep advancing the story wherever it leads," he added.

The White House this week was caught off-guard when Rudy Giuliani, now one of Trump's personal lawyers, told Fox News the president reimbursed his personal attorney Michael Cohen after he paid porn star Stormy Daniels $130,000 before the 2016 election through shell company Essential Consultants LLC in exchange for her silence about her alleged affair with Trump.

This appeared to contradict Trump's denial in April that he knew anything about the payment, but White House counselor Kellyanne Conway insisted during a separate interview Sunday that Trump's rebuff was in reference to knowing when the transaction originally took place.