The Celtic boss has been discussing the 29-year-old captain.
And the Norwegian says the midfielder is already a 'talented, motivated' coach who is taking notes and learning all the time as a player.
Indeed, when asked about Brown's chances of going into management in the future, Deila told reporters: "He’s loving Celtic and is a talented player but he’s a talented coach as well.
"He has leadership in him and he wants to develop as a coach. I think he can be a future manager. First, you need the motivation to be it and he has that.
"The next thing is learning and he has progressed that all his career as a footballer. He has the qualities to become a good manager and maybe I can pass the baton on to him."
Deila's predecessor at Celtic, of course, also captained the Bhoys, with Neil Lennon starring both as a player and manager at Parkhead.
Whether Brown has what it takes to lead the Scottish champions to similar success remains to be seen but the early signs have been encouraging.
The 29-year-old has worked with Celtic's youth team and development squad throughout this season, while team-mate Kris Commons revealed earlier this term that Brown has far more of a desire to go into coaching than he does at this stage in his career, despite being the younger of the two.
The Hoops finish their Scottish Premiership campaign at home to Inverness Caledonian Thistle on Sunday afternoon, when Brown will lift the league trophy with the rest of the Celtic squad.