The Express and Star reports that while the Baggies are yet to make a bid for the 26-year-old, Sean Dyche is aware of their interest in the player.
West Bromwich Albion manager Tony Pulis
Though the Express and Star adds that Mee is not the club's first choice target currently, he would surely be an excellent addition to Tony Pulis' squad, especially given their lack of depth on the left-hand side of the back four.
The former Manchester City trainee played every game last season to help Burnley clinch the Championship title and a place in the Premier League, and would bring a versatility that would make him incredibly useful either in the centre of defence or at left-back.
Burnley's Michael Keane and Ben Mee celebrate promotion
West Brom's chances of landing Mee could well depend on whether his centre-back partner Michael Keane remains at Turf Moor beyond this summer.
Leicester City have had a £12m bid rejected for Keane, with the Daily Mirror reporting on Wednesday that Dyche was open to selling the 23-year-old while the board were not.
And should the manager ultimately get his way if the Foxes come back with an improved bid, it seems unlikely that he will be quite as willing to allow Mee to follow him out the door for another of their Premier League rivals.