New Rangers boss Mark Warburton has already prised centre-back Rob Kiernan away from the Latics’ clutches this summer but the Scottish Championship side’s initial offer to take Tavernier to Govan has been rebuffed.
The 23-year-old has two years left on his current deal but spent the latter part of last season on loan with Bristol City as they cruised to the League One title.
Tavernier also enjoyed a loan spell with Rotherham, who are also keen on bringing the defender back to South Yorkshire.
Millers assistant manager Paul Raynor said: “James is a player who it is well documented that we like. Would we like him in the door? I think that’s quite obvious. He was fantastic for us. But he’s a Wigan player.
“I don’t know what has been going on behind the scenes if anything, but he’s certainly a player we’d like and love to have around.
“But he’s a Wigan player and that’s their prerogative. I know me and Steve (Evans, Rotherham manager) have spoke about how much we like him as a player and whether that has gone any further, I really don’t know.”