Former Gers boss Mark Warburton took charge at the City Ground last week and The Daily Mail claim he is ready to raid the Light Blues for one of his first signings.
Tavernier is believed to be on his summer shopping list, with the right-back position understood to be a real problem for Forest this season.
The 25-year-old is contracted to Rangers until 2019, having penned an extension last summer, but could be keen to move closer to his Yorkshire roots.
However, Warburton's successor, Pedro Caixinha, is expected to run the rule over his squad before making any decisions on the future of his players.
Tavernier was signed by Warburton two years ago from Wigan Athletic for a fee in the region of £250,000 along with Latics team-mate Martyn Waghorn, and has gone on to make 64 appearances in the league, scoring 11 times.