Reports say the Hammers, along with London rivals Crystal Palace, are interested in the youngster, who has failed to to make an impression at Arsenal but has done well out on loan abroad in recent years.
Campbell, born in Costa Rica, signed for the Gunners in 2011, but has struggled to force his way into the first team and instead was sent out to Lorient and then Real Betis on loan.
Betis are believed to want him again next season, even though he managed just two goals for them last term, but Hammers boss Sam Allardyce has also been alerted to his potential and could make a move.
Palace are believed to be keen, too, and a move to either of the Premier League clubs may well be more attractive to the youngster rather than another temporary switch abroad.
West Ham last week