REUTERS/Kevin Lamarque
In a pre-taped "60 Minutes" interview that aired Sunday, Obama dismissed Trump as a "publicity-seeker" capitalizing on "anti-immigrant" beliefs among Republicans.
"He's tapped into something that exists in the Republican Party that's real. I think there is genuine anti-immigrant sentiment in a large portion of at least Republican primary voters," Obama said.
Obama, pressed about Trump's chances, predicted that the Republican would ultimately flame out.
"I don't think he'll end up being the president of the United States," Obama said.
Trump responded both on Twitter (Obama was "terrible") and in a Monday morning "Fox & Friends" interview.
The business mogul took particular exception to Obama's focus on addressing global climate change.
"I don't get it: What's going on in our country? It can be fixed - and in many cases very easily - but we have somebody that's more worried about climate change than he is all of the problems that we have," he said, citing the economy and the Islamic State terror group. "All he talks about is climate change, and that is not our problem folks."
Trump also railed against both Obama's interview performance and the White House's support for the Trans-Pacific Partnership trade deal.
"I watched his performance last night and I thought it was terrible. I thought it was sad, because everything's negative. There's nothing positive. It's not like, 'How come are we doing so well here? How did you do these great trade deals?' Instead, we have a trade deal that's a disaster," Trump said.
He pointed out that former Secretary of State Hillary Clinton, the Democratic presidential front-runner, reversed her own position to oppose the deal last week.
"Hillary Clinton finally got smart, even though she supported it. She probably heard my stance," Trump suggested. "It's a horrible trade deal. It's a disgusting trade deal."
However, Trump was willing to give Obama a pass for predicting that he would ultimately lose the 2016 race.
"He can't say, 'Well, I think he's going to be the next president,' in all fairness," Trump said. "But I thought it was a little bit sad because there's so much negativity. It's just very negative."