The Baggies are believed to have offered a fee in the region of £5m but it is understood that Forest have told Albion that Antonio is not for sale.
The Reds value the 25-year-old, who scored 15 goals last season, much higher but would be unable to spend the proceeds from any sale while they adhere to the conditions of breaking Financial Fair Play rules.
West Brom boss Tony Pulis has been linked with a number of wide players, including Wigan’s James McClean and QPR’s Matty Phillips.
“We have to have a list of players and if people are unrealistic and ask too much, we have to move on and find people who are realistic and people who want to play for this club,” Pulis said earlier this week.
“At the moment the market out there has gone crazy and you have to buy into things but you have to be sensible as well.”
Antonio has been a long-term target for Pulis, but Forest chairman Fawaz Al Hasawi last week said that the Sky Bet Championship club had no intention of selling any of their key players.
He said on Twitter: “I would like to reassure our fans that I have no intention of selling Michail Antonio or any other player who wants to be part of this club.”
West Brom continued their pre-season preparations with a 3-1 defeat to Orlando City on Thursday morning. Pulis’ side will now face Charleston Battery on Friday in the second of three scheduled friendlies in the United States.