Sam Allardyce has already bolstered his attacking options by recruiting Ecuador international frontman Enner Valencia and former Lazio star Mauro Zarate this summer.
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Ferreyra is one of a host of South Americans at Shakhtar but, along with many others, has grown increasingly concerned at the ongoing violent unrest in eastern Ukraine.
The 23-year-old arrived from Velez Sarsfield last summer, where he starred alongside Zarate's brother, Roli, and it is thought his former club-mate has recommended him to West Ham.
The Hammers, though, are not the only side keen on Ferreyra as Newcastle, Sunderland and QPR are keeping tabs on his situation.
Shakhtar are reportedly considering the possibility of allowing Ferreyra to leave on loan with a view to a permanent deal, and a move to the Premier League is a natural attraction.
Ferreyra also has a European passport, thus meaning he would not need a work permit to play in England.