The Scotland midfielder has previously deputised for Brown and now takes the armband permanently, following the veteran’s departure for Aberdeen.
McGregor, a product of the club’s youth academy, will now lead Ange Postecoglou’s side into the new season as they look to make an impressoin at home and in Europe.
The 28-year-old said that he was delighted to take on the responsibility.
“To be the Celtic captain is a great honour for any player, and to be given the captaincy is a very proud moment for me and my family,” he said.
“I know I’m following in the footsteps of some great names in the club’s history, and I’ve been able to learn from one of the best in Scott Brown over the past few years.
“I’m delighted that the manager has put his trust in me by giving me the captain’s armband. It’s a new era for the club and I’m looking forward to playing my part in that.”
Postecoglou said: “Callum is a player who leads by example, both on and off the park, and he is a key figure in our squad.
“He has been at Celtic since he’s been a boy, he’s come through our Academy, had a loan spell away from the club to gain vital experience, and then returned to establish himself in the first-team. His qualities as a player are there for all to see, and that has been evident over the past few years.
“He knows what it takes to be a successful Celtic player, and he knows what it means to captain this club, and I’m delighted that he will be wearing the armband and leading the squad going forward.”
McGregor, who made his debut in 2014, has made more than 320 appearances for Celtic and has won six league titles, four Scottish Cups and four League ups.