Speaking on "Fox News Sunday," Priebus said Trump did not believe Putin's claim Friday that Russia had not attempted to influence the 2016 US presidential election.
"The President absolutely did not believe the denial of President Putin," Priebus said. "What the President did is he immediately came into the meeting, talked about Russian meddling in the US election, went after that issue at least two separate times."
Priebus said an "extensive portion" of the first meeting between the two leaders was dedicated to the issue and offered what he said was Trump's position.
"He's said they probably meddled in the election. They did meddle in the election," Priebus said. "The one thing that he also says -- which drives the media crazy, but it's an absolute fact -- is that others have as well. And that's true. China has, North Korea has, and they have consistently over many, many years."
Trump tweeted Sunday morning that he "strongly pressed" Putin twice on election meddling and that the two had discussed forming "an impenetrable Cyber Security unit" and a ceasefire in parts of Syria.
He also said, "Now it is time to move forward in working constructively with Russia!"
Priebus said Trump's tweets did not mean Russia was "off the hook."
"You can walk and chew gum at the same time," Priebus said. "We need to move forward and actually maybe work with Russia on bringing peace to Syria."
As for the areas of disagreement between the two countries over Syria, Priebus suggested the possibility of coming to an understanding, including on the position of Syrian President Bashar al-Assad, whom Russia supports and the US has said is a violent despot.
"It's yet to be seen what is going to be of Assad," Priebus said. "Certainly he is a butcher."
Former Director of National Intelligence James Clapper told CNN
on Thursday that he knew of no evidence to suggest other countries besides Russia had been involved in the election meddling. The US has in the past accused China and North Korea of hacking efforts in the US, saying in 2015 that China was likely behind a hack
of the Office of Personnel and Management, and the FBI said North Korea hacked
Sony Pictures in 2014.
Priebus' comments Sunday morning came as the latest assurance from the Trump administration that the President believes Russia attempted to influence the 2016 election, which was the conclusion of the US intelligence community.
UN Ambassador to the United Nations Nikki Haley said
on CNN's "State of the Union" that Trump was forceful in the meeting, and Russia was "trying to save face" in denying any election interference.
The meeting on Friday was between Trump, Putin, US Secretary of State Rex Tillerson and Russian Foreign Minister Sergey Lavrov. After the meeting was over, Tillerson and Lavrov offered different descriptions of the meeting on the election meddling issue. Tillerson said when Trump pressed Putin and he denied involvement, Trump moved on to other topics, while Lavrov suggested Trump had accepted Putin's denial.
Putin said as much
"I repeat, he asked a lot of questions on this matter," Putin said of Trump. "I answered as many as I could answer. I think he took it into consideration and agreed with it. But you should ask him what his opinion is on that."