The Argentine striker has agreed a season-long loan move to the London Stadium.
The Hammers confirmed that they have agreed a deal with Calleri and his club Deportivo Maldonado to bring the 22-year-old to England on a temporary basis.
That move appears to have been agreed despite reports of late interest from Spanish giants Barcelona, who hoped to wreck the deal.
In their report confirming the deal, the Guardian have claimed that the Hammers have had to shell out £4 million to ensure that they stayed ahead of their rivals to bring Calleri to London.
It is alleged that a personal agreement between the club's co-chairman David Sullivan and the owner of Stellar Group, who have ties to Maldonado, way key to making the move happen.
The paper also claims that Calleri sought the advice and approval of former Hammers and Argentina hero Carlos Tevez before making a decision on the move.
Calleri will become West Ham's seventh major signing of the summer and comes soon after the arrival of fellow forward Andre Ayew from Swansea City.
He has been with his country's Under-23 side at the Olympic Games in Rio de Janeiro until their exit from the competition this week, and will be hoping to force his way into the first-team picture under Bilic as soon as possible.