The Mirror says Sullivan has told West Ham he will only pay a knockdown fee for the club due to the Hammers' crippling debts of £100million.
Essex-based tycoon Sullivan, 60, is close to getting around £25m from Carson Yeung as part of the Hong Kong businessman's £80m takeover of Birmingham.
Sullivan is keen to reinvest the money in another football club and has already drawn up a list of possible sides to buy - with the Hammers at the top.
West Ham's owners CB Holdings, an off-shoot of Icelandic bank Straumur, which took over after the collapse of Landsbanki, are willing to sell.
If Sullivan fails to buy West Ham other targets include Norwich, Ipswich, Cardiff, Leicester and Charlton.