The 34-year-old, who netted a late winner for the Hoops as they recorded a last-gasp 2-1 win over Hamilton Accies at Parkhead on Wednesday night, told Sky Sports that he would "love to be manager", adding: "You need to learn.
"I've had some great managers - I've had Brendan [Rodgers], I've had Neil [Lennon], I've had Gordon [Strachan] and I've been learning a lot from [Celtic No.2] John Kennedy as well.
"I've been sitting down with people like that and learning a lot."
Brown has played for just two clubs during his professional career, leaving Hibs to join Celtic in the summer of 2007.
He was signed by Strachan, later playing international football under the former Leeds and Manchester United midfielder, and was a part of Celtic teams under Tony Mowbray, Ronny Deila, Lennon (twice) and Rodgers.
Brown's international career started under Walter Smith and he was subsequently called up by Alex McLeish, George Burley and Craig Levein as well as Strachan.
Speaking about Kennedy's influence, Brown added: "John's pretty much done it through every level at [Celtic] and now he's assistant manager, so he knows how it works at our club so I'd definitely love to give it a chance or a little shot and see how it goes."
Brown is contracted to Celtic until the summer of 2021 but has been tipped to enter coaching and management once he eventually hangs up his boots.
"I love football, I've been around it. I think when you retire you [sometimes] don't know what to do but still having that drive to Lennoxtown would put a smile on my face," he added.