Trump made the gaffe during an American Workforce Policy Advisory Board Meeting at the White House as Cook and the president's daughter Ivanka Trump were seated next to him.
Cook, who has sided with the Trump administration on certain issues, such as tax reform, but blasted them for others, including the trade war with China, appeared to have a bit of fun at the president's expense last week when he changed his name on Twitter to Tim Apple.
"I mean you've really put a big investment in our country, we appreciate it very much, Tim Apple", Trump said.
"Hey look, it's Tim Apple, son of Apple founder Steve Apple", said one user.
Mr. Cook has previously criticised Mr. Trump's decision to withdraw from the landmark Paris climate accord and condemned the President's travel ban.
"I just thought, why would you lie about that".
Cook took the stumble humorously, later changing his Twitter profile to Mr Tim with the Apple logo.
On Monday morning, Trump said he combined Cook's and the company's name together because it was faster. "It doesn't even matter!" During the conference, Cook sat behind a paper name tag that clearly read 'Tim Cook'. The president introduced defense company Lockheed Martin CEO Marillyn Hewson as "Marillyn Lockheed" at a conference previous year.