The 25-year-old has bagged 30 goals and 10 assists in the Championship this term, and he is also attracting interest from Tottenham Hotspur.
The Bees are looking to secure Premier League promotion via the play-offs and will cash in on Toney if they fail to.
West Ham, Chelsea and Spurs will have to part with £35 million to have a chance of landing the former Peterborough hitman.
Hammers boss David Moyes wants to sign another striker after failing to get a direct replacement for Sebastien Haller in January. It appears that the Brentford star has impressed him.
Leicester City and Everton have also registered an interest in Toney, but a move to the London Stadium could be more appealing, with regular playing chances almost guaranteed.
Hammers only primary striker Michail Antonio struggles with injuries, and Moyes knows he needs to get in another striker with European football almost guaranteed next season.
With more games to play, West Ham need a proven finisher like Toney and he could be an excellent acquisition for the club.
The former Newcastle United striker bagged 40 goals in 76 League One games for Peterborough United in two seasons before moving to the Championship.
He will be keen to play and prove himself in the English top-flight next season.