The Mirror says the Hammers had agreed a £6.7million fee with Estudiantes for the South American striker.
But the deal, including a four-year contract, is all subject to a work permit.
Watkins will now represent the club at a hearing next week to try to secure the 22-year-old's future in England.
Hammers boss Sam Allardyce wants the exciting Colombian to partner Andy Carroll.
Allardyce's scouts have been excited by Zapata's pace and power and want to land him.