Cruzeiro's director of football Alexandre Mattos has confirmed on the club's official website that terms have been agreed and that the player is on his way to London for a medical.
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"Cruzeiro is grateful to Wellington, and the great professional that he is," Mattos said.
"The players always honoured this club, and we like him very much. He has the club's doors open for the future."
Paulista has scored 19 goals in all competitions so far this campaign and will add to manager Sam Allardyce's striking options.
The Hammers have been busy in the transfer window so far having already signed Joe Cole from Liverpool and Marouane Chamakh from Arsenal.