West Brom say an "abhorrent message" was sent to Sawyers on social media during the loss to Manchester City.
They have called for "the toughest available legal punishment" and will ban the individual responsible from The Hawthorns for life.
"There is No Room For Racism. Anywhere. Challenge it, report it, change it," said the Premier League club.
"Albion will offer Romaine the relevant support required at this time and continues with its long-standing commitment to help rid football of racism."
West Brom boss Sam Allardyce said on Friday: "We dealt with it as quickly as we can. It leaves a big question about whether reporting it is enough.
"He is extremely upset, as anybody would be.
"It does become a greater thing to put right. But who does it lie with? Should a player take legal action? Social media sites need to act in a more responsible way. We have to do our best to cut this out."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!