The workout took place at Stanford, where the Colts were represented by general manager Ryan Grigson, coach Chuck Pagano, offensive coordinator Bruce Arians and quarterbacks coach Clyde Christensen, The Indianapolis Star reported.
The Colts didn't release details of the workout, but Luck's personal quarterback coach, George Whitfield Jr., told the newspaper that "things went well." Whitfield wasn't allowed in the private workout, however.
Owner Jim Irsay didn't attend the workout, according to the newspaper.
Irsay, however, mentioned the workout in a tweet on Tuesday morning and said the agent for Heisman Trophy winner Robert Griffin III denied the team's request to hold a private workout with his client.
"A Luck private workout 2day..we want 2b as thorough n disciplined n process of #1 pick..we wanted 2do private with RG3,but his agent said no," Irsay wrote.
The Washington Redskins, who hold the No. 2 overall pick, never requested and will not ask for a private workout with Griffin, believing they saw what they needed at his pro day, a team source told ESPN NFL Insider Adam Schefter on Tuesday.
The Redskins also will not request a private workout by Luck.
Griffin believes there's no need to risk an injury when he's assured of being selected in the top two spots, a league source told Schefter.
Irsay already has said he plans to take Peyton Manning's successor with the No. 1 pick, but no one within the organization is saying whether that will be the consensus favorite, Luck, or the fast riser, Griffin.
Luck is considered the most polished quarterback to come out of college since Manning was taken No. 1 overall by the Colts in 1998.
Manning's former offensive coordinator with the Colts, Tom Moore, told NBCSports.com that Luck is every bit the player scouts have tabbed him to be.
"He's the real deal. If you draft him you never have to worry about anything," Moore said in an interview posted Tuesday. "You know he'll be prepared. You know he'll be in shape. You know he'll study. You know he'll practice hard. You won't have to worry about anything. And you'll know he's going to go home and spend two-to-three more hours working on what you talked about."
Moore, according to NBCSports.com, was asked by Luck's father, Oliver Luck, to work with Andrew Luck in the classroom prior to the scouting combine in February.
"No. 1, he's very intelligent. He's extremely intelligent. Second, he's from a football family," Moore said, referring to the fact that Oliver Luck played four seasons as an NFL quarterback. "In the classroom, he has the things you're looking for. He has recall. You go over something on Monday, and on Friday when you put in the tapes he has instant recall. If I was still coaching I'd love to have him because he can do the things I like to do: audibles at the line of scrimmage, no-huddle. He has a great awareness of what's happening."
Moore said Luck is good enough to start in the NFL on "the first day of the first practice" and that he doesn't need a veteran to "mentor" him.
"I've never bought into the theory that you learn by watching. I think all that stuff is really overrated," Moore told NBCSports.com. "Is he going to make mistakes? Yeah, who hasn't? But I know Andrew's a strong enough person that he's going to handle the comparisons (to Manning), and he'll handle it perfect."
Information from The Associated Press was used in this report.