Calleri, previously linked with English pair Tottenham Hotspur and Brighton & Hove Albion, is currently on loan to Sao Paulo from Chilean side Deportivo Maldonado.
A group of investors were behind the 22-year-old's January transfer from Boca Juniors, and his future is said to lie in their hands.
An agreement with Inter Milan had been reported, but Lance states that the investors' preference is a move to England.
West Ham are known to be seeking attacking reinforcements this summer, and have seen big-money bids for Lyon's Alexandre Lacazette and Bournemouth's Callum Wilson turned down this week.
Negotiations with the Hammers are said to be at an advanced stage, and far closer to those with Inter, which will come as a disappointment to the player, who favours a move to Italy where his family have roots.
Sao Paulo face Atletico Mineiro in the quarter-finals of the Copa Liberatdores - a competition in which Calleri has scored eight goals in nine appearances so far - in the early hours of Thursday morning, and their progression to the last four would see the player's stay in Brazil automatically extended until July.
If not, Calleri will likely return to his parent club at the end of June, with a move to Europe expected to follow.