Tavernier, 23, has been on the books with Leeds United and Newcastle during his career, but is currently in League One with Wigan Athletic following their relegation from the Championship at the end of last season.
The Latics only signed Tavernier from Newcastle last summer, but he struggled to make an impact at the DW Stadium, and was sent out on loan to Bristol City for the second half of last season.
Now back with Gary Caldwell's Wigan, it seems that the full back could be on the move again - if Rangers get their way, that is.
STV reported on Thursday that Rangers had seen a bid for Tavernier rejected, with new manager Mark Warburton keen to bring in a new right back this summer.
With that in mind, The Scottish Sun suggest that Rangers are now set to return with a new and improved bid for the defender within the next 48 hours - showing just how serious Warburton is about signing the right back.
What does this mean for Wabara?
Rangers have been keen to bring in new signings all summer, and their pursuit has led to them taking a number of players on trial.
One of those is defender Reece Wabara, who is a free agent having left Doncaster Rovers earlier this summer. The right back has been looking to impress Warburton, but was intriguingly left out of Rangers' 2-0 friendly win over Ayr United on Wednesday.
The heightened bids to land Tavernier could suggest that Wabara isn't in Warburton's immediate plans at Rangers, and could be back looking for a new club imminently.