McClean has been linked throughout the summer with a move to Stoke, whose manager Gary Rowett tried to bring the winger to Derby County in January.
Should he complete his switch to the Bet 365 Stadium, McClean will become the latest player to depart West Brom following the Baggies' relegation from the Premier League.
And the Republic of Ireland international, who recently entered the final year of his contract, could be followed out of the Hawthorns by the former Stoke goalkeeper Ben Foster.
According to the Telegraph, Foster is on the verge of joining another of his old clubs, Watford, in a £2.5 million deal.
Stoke and West Brom are expected to compete for the Championship title this coming campaign.