The 32-year-old has earned legendary status during his nine-year stay at Ibrox and has won five major domestic honours, including the 2020-21 Scottish Premiership title and last season's Scottish League Cup.
Operating as an attack-minded right-back, Tavernier has scored and remarkable 125 goals and registered 129 assists in 460 appearances for Rangers across all competitions, netting at least 18 goals in each of the last four seasons.
The Bradford-born defender still has two years remaining on his contract with the Glasgow-based club, but speculation over his long-term future has been rife in recent weeks, with manager Philippe Clement in the process of reshaping his first-team squad.
While Rangers have welcomed seven new arrivals so far this summer, eight players have departed Ibrox and Tavernier could be the latest name to leave the club ahead of the new season.
Trabzonspor 'hopeful' of finalising Tavernier signing
Last month, it was reported that Tavernier had accepted an 'astronomical' financial package to join Saudi Pro League club Al Ettifaq and reunite with former Rangers manager Steven Gerrard, but those rumours have since cooled.
Multiple reports, including one from Sky Sports News, have since claimed that Turkish Super Lig outfit Trabzonspor have seen an initial offer for Taverner rejected by Rangers, but they are expected to return with an improved bid.
According to TEAMtalk, Rangers are willing to sell their skipper this summer in order to fund new arrivals, and Trabzonspor are currently just £1m shy of the Gers' asking price.
Trabzonspor are said to have 'real hope' of striking a deal with Rangers to sign Tavernier, who has been promised a 'very big' wage rise on a two-year contract which includes an optional one-year extension.
Tavernier could link up with former teammates John Lundstram and Borna Barisic after they both left Rangers upon the expiration of their contracts at the end of June to join the Turkish club on free transfers.
Trabzonspor president confirms imminent right-back signing
Trabzonspor appear to be pushing for the signature of Tavernier, with the club's president Ertugrul Dogan confirming that a right-back arrival is imminent.
"Our plan B (after Thomas Meunier's departure) was ready and we took action immediately. Our new right-back will arrive in a day or two," Dogan told reporters as quoted by the Daily Record.
"Our only thought is to create a team that makes the fans happy, stands tall on the field and fights. Our fans should not worry. Trabzonspor will fight for the top.
"Today or tomorrow, Trabzonspor will complete the right-back transfer. We are planning to make five to six more transfers. We aim to complete two to three of them in the next two to three days."
Tavernier is not the only high-profile player who could be departing Rangers this summer, as both Connor Goldson and Scott Wright are believed to be the subject of interest from clubs in England.