The former Juventus centre-forward is wanted by the Hammers ahead of the January market kicking off.
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Globo Esporte says the Hammers have tabled a loan-to-buy offer for Jorge, with the permanent option worth €25m.
However, Cruzeiro have instantly rejected the attempt, insisting Jorge isn't for sale.
Despite Cruzeiro's stand, West Ham intermediaries believe they will sell for a higher bid.
Jorge, 23, scored a league-leading 21 goals in Brazil's Serie A in 2025.