Former Newcastle player Tavernier was signed by Mark Warburton from Wigan in the summer of 2015, when Rangers were in the Scottish Championship.
He helped the Ibrox side win that division and captained the side in the absence of the injured Lee Wallace as Warburton made way for Pedro Caixinha.
When Steven Gerrard arrived as Rangers manager in 2018, Tavernier became captain full-time as Wallace’s contract ran down and was pivotal part of the Gerrard title-winning team, Giovanni van Bronckhorst’s Europa League final and Scottish Cup-winning side and Philippe Clement’s League Cup victors.
Tavernier also wore the armband during Michael Beale, Barry Ferguson and Russell Martin’s brief stints as boss. This season under Danny Rohl since October, the defender has not been an automatic starter, with Dujon Sterling often preferred at right-back.
Jack Butland and Nico Raskin have deputised as captain, with both still under contract next season.
“We’ve shared the highs and the lows together and some truly incredible moments along the way,” Tavernier said.
“To all the players I’ve shared the pitch and dressing room with over the years, thank you. The bonds that we’ve built, the battles we’ve been through and the memories we’ve created are what make this game so special.
“To the staff who work tirelessly behind the scenes every single day, thank you for everything you’ve done for me and the for the team. Your support often goes unseen, but it has never gone unappreciated.
“And to the fans, your support home and away, through every high and every difficult moment has meant more than I can ever fully express. You are the heart and soul of this club and it has been a privilege to play in front of you.”
Tavernier added “this isn’t the end just yet”, without confirming where he would be next season.
Second-placed Rangers are in a three-way fight to win the Scottish Premiership, with Hearts one point ahead and Celtic two points off the Ibrox side and five games to play.
“There is still a season to finish and we are fully focused on pushing for the league title,” Tavernier said. “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.”