Centre-back Simunovic, who arrived at Celtic from Dinamo Zagreb in 2015, has committed his future to the champions with a four-year contract that runs until 2021.
Both players are set to feature for the Hoops when they open their Champions League campaign against Paris Saint-Germain at Parkhead on Tuesday.
Brown has already made 11 appearances for Celtic this season and has been crucial in the team reaching the group stages of the Champions League for a second consecutive season.
The 55-times international has made 427 appearances for the Hoops, scoring 37 goals and supplying 55 assists.
Simunovic has played in eight of Celtic’s 11 matches this season, with injury ruling him out of three of them.
He is widely regarded as the club’s outstanding centre-back but has struggled with injury problems since his move from Croatia.
In other Celtic team news, the club are in advanced talks to sign up an enforcer, with the deal to be done this month.
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