"It's given me so much, both on and off the pitch, and from the moment I arrived I've always tried to give everything I had to represent it in the right way.
"To have had the honour of being made captain of this club since 2018 is something I will always be incredibly proud of.
"Leading this team out, wearing the armband and being trusted in that role has meant more to me that I can properly put into words."
Second-placed Rangers are in a three-way fight to win the Premiership, with Hearts one point ahead and defending champions Celtic two points off the Ibrox side with five games to play.
"There is still a season to finish and we are fully focused on pushing for the league title," Tavernier added. "I will continue to give absolutely everything, as I always have until the very last day.
"My focus right now is on finishing this season the right way, together."
He veteran added that "this isn't the end just yet", without confirming where he will play next season.