The Premier League new boys are considering a summer swoop for the Welshman.
While the Mail previously downplayed any speculation linking the Welshman with a potential return to Aston Villa where he came through the youth ranks, it appears that Tony Pulis will still have offers to respond to with the Clarets monitoring his situation.
The 33-year-old is about to enter the final year of his contract at the Hawthorns, and the Mail suggests that that may tempt the Baggies into selling the stopper this summer.
With Tom Heaton enjoying another excellent season at Turf Moor, it is difficult to imagine that joining Sean Dyche's side will see Myhill be anything more than the England keeper's understudy.
But with only 36-year-old Paul Robinson currently providing back-up to Heaton, Dyche will know that it would be a gamble to go into the new campaign without bolstering his options between the sticks.
And if Dyche is able to promise the amount of first-team football that Pulis cannot, Myhill may decide that he is at the stage of his career when he cannot reject such an offer.