The Wales international made 43 league appearances last season as Vincent Kompany's side cruised to the Championship title.
Roberts went on to start eight of Burnley's first nine Premier League games this season, but he has since lost his place in the lineup.
Since making his most recent top-flight start against Brentford in October, Roberts has had to settle for just four substitute appearances in the Premier League.
Instead of utilising the 28-year-old, Kompany has preferred to use summer signing Vitinho as his first-choice right-back in recent months.
With Roberts struggling to gain regular game time, he is now being linked with a move away from Turf Moor this month.
A recent report claimed Championship high-flyers Leeds United were showing an interest in the defender to Elland Road.
However, according to the Daily Mail, the Yorkshire club are facing competition from Premier League side, Brentford.
The Bees have already brought in Tottenham Hotspur loanee Sergio Reguilon, who can play as a left full-back or further up the pitch in the wing-back role.
With Aaron Hickey still sidelined through injury, Brentford are now looking to strengthen the opposite side of the pitch.
Thomas Frank's side have already been active in the current transfer window, having recruited Yunus Emre Konak and Hakon Valdimarsson as well as Reguilon.
They are also on the verge of signing Antonio Nusa after reaching an agreement with Club Brugge over a deal that will see the Norwegian winger return to Belgium on loan until the end of the season.
While the Bees have joined Leeds in the race for the Roberts's signature, Feyenoord are another side keeping tabs on the player's situation.
The Dutch giants are on the lookout for a new full-back after recently failing in their efforts to sign the Welshman's compatriot Neco Williams from Nottingham Forest.
Roberts, who has 18 months left on his current contract, has made a total of 87 appearances for Burnley since arriving from Swansea City in the summer of 2021.