The Hammers have indicated they want a prolific goalscorer to spearhead their push for Champions League football.
Ahead of their move to the Olympic Stadium, West Ham have been linked with the likes of Alexandre Lacazette, Christian Benteke, Jamie Vardy and Michy Batshuayi.
However, reports have suggested a £31million bid for France international Lacazette has been rejected by Lyon, leaving Sullivan somewhat frustrated.
The West Ham chief accepts he will have to splash the cash though to land a dream forward.
Juventus forward Simone Zaza is edging towards a summer exit with Chelsea and Southampton also keeping tabs on the Italian.
West Ham are keen on Zaza, and Italian outlet Corriere Della Sera claim the Serie A giants - due to play against the Hammers in August - value the striker at £19m.
With Lyon pricing West Ham out of a move for key target Lacazette, a swoop for 24-year-old Zaza could still prove lucrative.
So far, centre-midfielder Havard Nordtveit is the east London’s club’s only summer addition, but Sullivan insists more are to come.