Following the Baggies' relegation to the Championship in May, Dawson has been attracting attention from top-flight clubs over the summer, with the Midlands club, so far, rejecting bids from West Ham and Burnley during this transfer window.
Leicester City and Celtic have reportedly also expressed an interest in signing the central defender, who has been at the Hawthorns for seven seasons.
The defender was part of a trio of players who refused to take part in the club's pre-season tour of Portugal earlier this summer - a group which included winger James McClean and goalkeeper Ben Foster.
Both McClean and Foster were subsequently sold to Stoke City and Watford respectively, and Dawson is expected to follow them out of the Hawthorns ahead of the Baggies' first season in the second tier of English football since the 2009/10 campaign.
The centre-back joined Albion in 2011 from Rochdale and has made 180 appearances in all competitions for the Midlands club. He appeared 28 times in the league for the West Brom last season as the club were relegated from the Premier League to the Championship.
Despite missing the Portugal trip, Dawson has featured for the Baggies during their pre-season campaign. He was part of the West Brom squad which travelled to St. Andrews for a training camp in Scotland, but was the only unused substitute during their one game north of the border against Aberdeen, causing further speculation about his future.
However, the 28-year-old did feature for Darren Moore's squad in their 3-2 friendly win over Barnsley last Tuesday, playing the first 45 minutes for West Brom before being replaced by fellow defender Ahmed Hegazi at half time.
Manager Moore said explained that Dawson's absence from training was due to a thigh injury, but the Mirror have revealed there is a deeper reason for the centre-back missing the squad's Championship preparations.