Ex-skipper Boyd feels that with the current situation of Old Firm rivals Rangers winning the past three Scottish titles, there was very little chance Brown would leave the club high and dry.
"Scott might feel he has unfinished business with Celtic. He's lifted the league championship before but not as captain,” Boyd said.
"You wouldn't want Rangers getting close to nine or ten again.
"The longer a run like that goes on, the team from the other side of town have the bragging rights as they're the one winning the championships.
"The sooner you can put a stop to that the better for the Celtic supporters. Scott will want to play his part in that I'm sure. He has had a few injuries and the like but him moving out to the right to accommodate Beram Kayal has made the midfield a lot more solid.
"Towards the end of the season he put in a lot of good performances.
"The midfield have put in some fantastic performances and he's been a part of that.
"To have someone with that drive and determination can always fit into the Celtic team."