In a tweet, Trump also boasted: "I've made a lot of bankers, and others, look much smarter than they are with my great economic policy!"
Dimon made his impromptu comments at an event at the bank's Park Avenue headquarters in New York.
"I think I could beat Trump," Dimon said. "Because I'm as tough as he is, I'm smarter than he is. I would be fine. He could punch me all he wants, it wouldn't work with me. I'd fight right back."
The CEO walked back the comments about an hour later.
"I should not have said it. I'm not running for president," Dimon, 62, said in a statement.
Trump shot back Thursday morning on Twitter.
"The problem with banker Jamie Dimon running for President is that he doesn't have the aptitude or 'smarts' & is a poor public speaker & nervous mess," Trump said.
The J.P. Morgan leader has in the past made strong comments about the political situation in the U.S., but typically directed his ire at partisan gridlock and not Trump.
Last year, Dimon said it was "almost an embarrassment to be an American citizen traveling around the world and listening to the stupid s--- we have to deal with in this country."
-- With reporting by CNBC's Hugh Son