McClean has made 112 appearances for the Baggies since signing from Wigan, scoring just five goals.
The 29-year-old found game time hard to come by under both Tony Pulis and Alan Pardew, but started the last six games of the 2017/18 season under Darren Moore as West Brom almost pulled off an unbelievable escape.
McClean is entering the final year of his four-year contract with the Baggies, whilst the club reportedly turned down a £4m bid from Stoke for the player's services earlier this month.
Moore is looking for a fee in the region of £6m for the winger who cost £1.5m back in 2015.
The Potters have already completed the signing of Benik Afobe from Wolves as well as the signing of Nigerian midfielder Karo Etebo.
Stoke City fans will be concerned at the prospect of a 29-year-old in decline arriving at the club for a fair bit of money, especially given the divide that the player causes amongst fans for his political stances as well as his playing style.