Osorio plays for Chilean top-flight side Universidad de Chile and has just won his third cap for the national team.
According to The Sun, West Ham are expected to table an offer worth £7.5million for the 19-year-old.
A winger by trade, Osorio has already turned down a move to Brazil because he wants to play in Europe.
AC Milan are also keen on a move as they propose the idea of purchasing the player and loaning him out.
Universidad will sell the starlet at the right price with West Ham keen to act quickly to get a deal done, with Wolves also interested.
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This is set to be completed in the coming days when the financial package is agreed upon.