Trabzonspor pursued the 32-year-old player, with interest also coming from clubs in Qatar and Saudi Arabia. Despite this, no offers were received that matched the valuation set by the Ibrox club for the talented right-back.
Speaking to ahead of Sunday's Premiership match against Hibernian - live on from Ibrox - he said: "There was always going to be reports and links. I am in contract until the end of next season.
"Rangers have always given me that platform to play my football and the foundations to do something that a lot of people would love to do, so I've always got a great debt to pay, but it is football.
"If teams come in, they come in, but I am fully focused on the job in hand and I know I've still got a lot to achieve."
TrendingTavernier - now in his 10th season at Ibrox - has won a Scottish Premiership, Scottish Cup and League Cup title, plus helped Rangers reach the Europa League final in 2022.
He has also scored 126 goals and contributed 132 assists during his time in Glasgow, becoming the highest-scoring British defender of all time.
However, he has faced criticism since joining in 2015 - most recently after Rangers' 3-0 defeat at Celtic Park, as fans waited outside Ibrox for players to return so they could express their frustration over performances and results.
"It is part of the parcel with football," Tavernier told . "You see it up and down the country.
"As fans they want to display their frustration, you can understand the frustration that people have and obviously, us as players are frustrated as well.
"We never go out there wanting to lose a game, or even draw a game - especially playing for Rangers we always want to win. We will always give our 100 per cent effort to that cause.
"On certain days it can go too far but it is something in the past, it is not something I dwell on. I always keep looking forward and obviously, it motivates me more to do better."
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