The Hammers have lined up a deal with Argentine side Estudiantes for the highly-rated 22-year-old.
Zapata is eager to seal a move to the Premier League, but his application will have to go to appeal and West Ham are working on their case.
A number of high-profile figures in world football are set to give references about Zapata, who is valued at close to £8million, and so West Ham are hopeful they will get a permit on the 'special talent' avenue.
The fact Colombia are now ranked in the world's top three is also likely to be taken into account and, although Zapata is yet to win his first cap, he does look set to become part of their squad before next summer's World Cup finals.
A source close to the deal confirmed to Sky Sports that he was 'hopeful' of securing what he believes would be a 'dream move'.