Steven Gerrard, however, has rubbished those rumours, stating the Englishman is going nowhere.
It was reported that the club have been looking at Leonel Di Placido, an Argentine full-back who is wanted by teams in Europe.
Playing for Lanus in Argentina, the player has been interesting Crystal Palace and Spanish sides. It was understood that Rangers saw him as a possible replacement for Tavenier.
But that is not the case.
"I read that we were linked with a right back which is total rubbish,” Gerrard said.
“I’ve said on record before that James is a big part of what we are doing here and he has been one of my most consistent players as well."
The Rangers captain had previously linked with a move to Newcastle United this summer.
Gerrard was responding to questions regarding the captaincy having been effusive in his praise of Ryan Jack.
The centre midfielder was impressive in the win over Hibs when it seemed that he could lose his place with the signings of Joe Aribo and Steven Davis.
However, the former Aberdeen captain is a big-game player, according to Gerrard, who also confirmed that he and Jermain Defoe could be in line for new deals.
The striker, who hit a hat-trick against Hibs, is on loan from Bournemouth with his contract expiring at he end of the season.
He said: “If Jermain stays in the form he is in, Jermain has a conversation with me at some stage further on in the season.
"I will reward players who reward me with performances.
“The main ingredient why Ryan is where he is is because he has got something inside him. It’s called a big heart.
“Ryan has given me every drop, everything inside his body. At the right time, I am sure he will be rewarded.”