This was Trump's first time before reporters since his public, freewheeling immigration negotiation with Republican and Democratic lawmakers on Tuesday.
"We'll see what happens", Trump said.
Judiciary Committee Chairman Charles Grassley, an Iowa Republican, objected to Feinstein's release of the testimony.
"If I were his lawyer, I'd say, 'Keep your mouth shut and just talk to me or you'll end up like Bill Clinton, '" Jarrett said on "Lou Dobbs Tonight".
President Trump refused to say Wednesday whether he would grant an interview to special counsel Robert Mueller, who is probing Russian interference in the 2016 presidential election and possible ties between the Trump campaign and the Russians who attacked that election.
Trump's attorneys were said to have asked whether it could be in the written format. "Putin can't love that", he said.
"So we can conceivably go back in", added the president, who announced his intention to pull out of the 2015 accord on curbing global warming emissions last June. Washington will continue to "secure critical technologies" while "maintaining America's military edge" over near-peer competitors such as Russian Federation and China, the source said.
"The salvo from Mr. Trump, who has long expressed hostility toward traditional press freedoms, followed a days-long effort by him and his team to undercut the unflattering portrayal of the White House in a new book by the writer Michael Wolff".
Screengrab via IGN Walkthroughs YouTube The F-52 as seen in the game “Call of Duty Advanced Warfare”
Mueller has also tentatively set a trial date in May for former Trump campaign manager Paul Manafort, who was indicted in October on charges, including money-laundering conspiracy, related to his lobbying work on behalf of a Russia-friendly Ukrainian political party. The closest he came to addressing the Mueller interview question was one of his preferred replies: "So, we'll see what happens".
Trump went on to blast allegations that his campaign colluded with Russian Federation, arguing that such claims have hurt his presidency.
Far more than any of his predecessors, Mr. Trump likes to talk up publicly the quality of the US defense products and the need for foreign militaries to consider buying American, as he did in a bilateral meeting with visiting Norwegian Prime Minister Erna Solberg in the Oval Office on Wednesday.
"It's not because all Norwegians are rich", she said in the interview.
The report tracks Russian efforts in 19 countries, chronicling misinformation campaigns, the funding of far-right political causes and the manipulation of energy supplies long before 2016 in an attempt to glean lessons for USA officials considering how to counteract similar efforts here. He remained an ally of Trump through the fall but their relationship fractured following the publication of "Fire and Fury".
USA intelligence agencies assessed past year that Russian Federation had conducted a campaign targeting the election with the goal of hurting Clinton's chances of winning while boosting Trump.
Trump tweeted Friday amid criticism over his comments during a White House meeting Wednesday.
Trump made the remark Thursday during a White House meeting after senators discussed revamping immigration rules.