Tavernier has captained Rangers for the past eight years, winning the Scottish Premiership in 2021, the Scottish Cup the following year and the League Cup in 2023.
He also skippered Rangers in the 2022 Europa League final against Eintracht Frankfurt, suffering the disappointment of defeat in a penalty shootout.
Rangers trail leaders Hearts by a point this season and will hope to overhaul the Edinburgh club to give Tavernier another Premiership winner's medal before he leaves.
The 34-year-old, whose contract expires at the end of the season, said on Instagram that leaving "hasn't been an easy decision".
"This club has been a huge part of my life and my family's over the past 11 years," he said.
"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."
The English right-back is the highest-scoring defender in British football, having amassed 144 goals in 562 Rangers appearances and 152 goals overall.
He arrived from Newcastle in 2015 when Rangers were in the Scottish Championship, having originally been demoted to the fourth tier after the club went into administration.
Tavernier helped Rangers win that division and when Steven Gerrard became Rangers manager in 2018, he made him captain full-time.