Per the source, the Hammers are plotting a January move for the Portuguese international.
Bruma has hit the ground running this season and has contributed to seven goals in his last eight games – four goals and three assists.
The young winger is currently one of the most highly rated players in the league and has been arguably Galatasaray’s player of the season.
The Lions do now however, want to sell Bruma for under £20 million and want to keep hold of him until the end of the season.
Galatasaray are currently in talks with Bruma over extending his contract and including a buyout clause option that reflects their valuation.