McClean could be pulling on a Stoke City shirt next season, as the club have reportedly improved on their opening offer of £4million.
The Republic of Ireland international, who has only a year left on his contract at the Hawthorns, has struggled for first-team action in a season which saw the 'Baggies' relegated from the Premier League.
If the offer goes through, it would mark the end of a three-year period at the club, after joining from Sunderland in 2015.
His acquisition would continue Stoke's competitiveness in this summer's transfer market as they look to secure promotion back to England's top flight.