The 35-year-old is eager to leave the Hawthorns this summer following the club's relegation, and is said to be growing frustrated at Albion's firm stance over offers thus far, which seemingly led to his refusal to join the club's training camp in Portugal on Tuesday.
Foster is due to be disciplined by the Baggies for their absence at compulsory pre-season training, as he looks to force their way out this summer.
According to the Daily Mail, West Brom went as far as laughing out the Hornets' £1m bid as they did not deem the offer a fair reflection of Foster's importance and value to the club.
The Baggies had expected an enticing bid from the Premier League outfit for a keeper who has attracted attention from a handful of clubs following a consistent season in between the sticks, despite their relegation to the Championship.
Foster has spent seven seasons at the Hawthorns and is contracted until 2019, and his recent reluctance to stay the course has come as a surprise to the Baggies as he had vowed to be a part of the club's fight to return to the top flight just two months ago.
Foster said, in April: "'I'll be here. I've got no interest in trying to move clubs. I'm very settled and I'm sure there are quite a few others who would love to stay and help us get back to the Premier League."
Dawson, meanwhile, is expected to be allowed to leave once his valuation is met after being the subject of £12m bids from both West Ham and Burnley.