Warburton, who was appointed last month, has already snapped up the likes of Wes Foderingham, Danny Wilson and Rob Kiernan, but needs to continue to add to his squad this summer.
Having released Sebastien Faure and Richard Foster, Rangers are in desperate need of at least one right back, and whilst former Manchester City youngster Reece Wabara is on trial at Ibrox, it seems that Warburton has his eye on another potential signing.
STV report that Rangers have made a bid for Wigan defender James Tavernier, but the Latics have rejected that offer, in search of a higher fee.
Sky Sports also state that the bid is just an initial offer from Rangers, suggesting that Warburton remains interest in a deal to bring the former Newcastle United full back to Glasgow - budget permitting.
What would Tavernier offer Rangers?
The 23-year-old is a pacey right back who loves to get forward, displaying his quality with Rotherham United during the 2013/14 season.
Tavernier has had a frustrating last year with Wigan and Bristol City, but he has plenty of potential - something Warburton is no doubt keen to tap into.
The former Leeds youngster would be an excellent signing for Rangers, but they must be wary of how much they spend on the full back, considering that they still need to strengthen other areas too.