Virginia Democratic Senator Mark Warner says Democrats ought to continue to press the urgency to the American people that Russia will continue to influence the U.S. election process, saying every American should take former special counsel Robert Mueller's testimony on Capitol Hill as a "warning."

"Where we've been focused is the point that Bob Mueller made repeatedly yesterday, not only did Russia attack us in 2016, but they will be back in 2020...I've got no interest in re-litigating 2016, I am interested in making sure that we have appropriate protections in 2020," Warner told "CBS This Morning" on Thursday.

During Mueller's long day of congressional testimony, the former FBI director said "many more countries" are developing the capability to do what the Russians did in 2016. Mueller said he expects Russia to continue to interfere in the next election, saying, "they're doing it as we sit here."

"I would hope that every American, I don't care where you fall on the partisan scale, would take that as a warning and recognize that same message is coming out of Donald Trump's own director of the FBI, Chris Wray, it's coming out of Trump's own Director of National Intelligence Dan Coats, and this ought to be a national security issue," Warner said.

Warner said that it's on the White House and Senate Majority Leader Mitch McConnell to bring up comprehensive legislation to address similar foreign influences. He noted, however, that Mr. Trump doesn't seem so interested.

"What bothers me is this president seems so obsessed about his own personal well-being that he dismisses this threat and frankly when he goes to a G-20 meeting and makes light of this issue with Vladimir Putin, he does not make our nation safer," he said.

Mueller on Wednesday insisted the "ability of our intelligence agencies to work together in this arena is perhaps more important" than developing specific techniques.

"Any legislation that can encourage agencies to work together to fight foreign interference should be pursued soon and aggressively, Mueller urged.

Warner, who serves as vice chairman of the Senate Intelligence Committee, says that while his committee is still in the process of finishing its own probe into Russian interference, they still haven't received access to "reams of counterintelligence evidence."

"[Mueller] said they didn't even look at collusion because collusion is not a legal terminology, they did say there were well over 100 contacts between Russians and individuals in the Trump campaign," Warner said of Mueller's testimony.

He added, "That's just not normal."