The Hammers are interested in the QPR attacker but face stern competition from Crystal Palace for his signature while Leeds and West Bromwich Albion are also keen.
Moyes and the Hammers board are big fans of Eze, Moyes scouted the youngster last season himself and while defence is the priority this summer, an attacker is still a possibility to bring in.
West Ham will have to pay at least £16million to prize Eze from Queens Park Rangers, which means they have to allow some players to leave.
West Ham is reportedly ready to listen to offer for Anderson, Manuel Lanzini, and Jack Wilshere in order to raise funds for the England youth international.
West Ham and Moyes want to build a young and hungry squad next season and Eze has been identified as one of the players to start the revolution at the London stadium.
