The Premier League club have submitted a bid for the 26-year-old, with discussions over a possible transfer now under way.
It remains to be seen whether the move will go through. Anthony joined Burnley on loan in the summer of 2024 before the move was made permanent.
Since then, the Englishman has grown into an important player for the club. Burnley finished 19th in the Premier League last season and will play Championship football next term.
Anthony still enjoyed a productive campaign, scoring eight goals and providing two assists in 37 league appearances, 32 of them starts.
He is under contract with Burnley until the summer of 2029. Brentford finished ninth in the Premier League last season.
Anthony has previously played for Leeds United and Bournemouth earlier in his career.