The allegations come after a week of big developments in the nation's capital, including the firing of Comey, who was investigating the ties between the Trump campaign and Russia, and reports that Trump revealed highly classified information to Russians in a closed-door meeting.
Kovacevich said while Trump had success on the campaign trail, presiding over a bureaucracy and the world's economies is a completely different process.
"His natural personality helped him in the campaign, and his natural personality is hurting him in becoming a great president," he said.
Kovacevich's comments followed those made by former GE CEO Jack Welch earlier Wednesday, who said Trump has "crappy management practices."
"I give him a D minus, and I'm being an easy grader," Welch said on "Squawk Box."
Don Peebles, CEO of The Peebles Corporation, said newcomers to politics always have a difficult time making the transition.
"What's missing here, I think, is there is no stabilizing veteran of Washington, D.C., who has the respect of the president who can help him shape his message," Peebles told "Closing Bell."
"His communications office has served him poorly right now."
Kovacevich agrees Trump needs to surround himself with a good team.
"In general, I think he appointed some very good people, but I think the ones that aren't working out, he's going to have to make some further changes," he said.
—CNBC's Matt Belvedere contributed to this report.