The two London clubs are prepared to offer as much as £2million for the speedy England Under-17 striker's services in the close season.
Reports claim Arsenal are ready to pay £1million but Chelsea could double that in an attempt to wrap up a quick deal.
Chelsea will also hope the fact that former Blues defender Steve Clarke is manager of West Brom will swing a deal in their favour as Romelu Lukaku has spent this season on loan at the Hawthorns which could also help.
Arsene Wenger will not give up without a fight and his reputation for developing young players such as Theo Walcott and Alex Oxlade-Chamberlain could tempt Brown.
Clarke is still hoping to convince the player to sign pro terms with the club as soon as he turns 17, but interest from two of the Premier League's big guns could turn his head.