The Northern Ireland international has rediscovered his best form at Ibrox since his loan move from Fulham was made permanent last year.
Davis could be tempted by a return to the Premier League after winning the SPL title with the Gers last season.
Rangers chief Walter Smith will beek keen to keep Davis, but a bid of around £5million might persuade Smith to allow the former Aston Villa star to leave.
The 24-year-old started his career at Villa Park before moving to Fulham for a feein the region of £4million.
Things didn't work out for Davis at Craven Cottage and the hard-working midfielder moved to Rangers on loan before agreeing a four-year contract.
Black Cats boss Bruce is eager to bolster his midfield ahead of his debut campaign at the Stadium of Light.
Source: Team_Talk