The former Wigan Athletic man has been in scintillating form this season.
Warburton and Tavernier took the Scottish Championship manager and player of the month awards for August, in what is a measure of the Gers' performances so far this season.
“For a start it is always about the team, their work and then getting rewards for their work and their commitment, any award we get is a testament to all their endeavours,” Warburton told Rangers.co.uk.
“I’m delighted for Tav but he would be the first to say it’s about the team but he has worked so hard and he has really found a home here.
“He’s moved around a lot but he loves Glasgow, he loves the club, he is well respected in the dressing room and hopefully he will be here for a long time,” he added.
The Rangers right-back is well travelled despite his tender years and has seen action at no less than 10 clubs already.
After coming through the Leeds United academy, Tavernier was picked up by Newcastle United, but made just two senior appearances in his five years at the club.
His stay on Tyneside was punctuated with six loan spells to various Football League clubs before a permanent move to Wigan was made last summer.
Tavernier’s stint at the DW Stadium was short-lived and he was sent out on loan to Bristol City in January.
The pacey full-back appears to have finally settled at Ibrox and, like Warburton, Rangers fans will be hoping he sticks around for a good while yet.