The Hammers boss wants to bring in Zapata, a 22-year-old Colombian striker with Argentine outfit Estudiantes, but the move has been held up by work permit issues.
"It's whether we think we can get the work permit through, that's the only thing," Allardyce told Sky Sports.
"Based on the fact that he's not a full international, we would have to go to a hearing.
"We'd have to decide in the next 24 hours whether - on our part or the player's part - that would be worth doing."
One player Allardyce was happy to rule out was West Brom's Dorrans, after being linked with a move for the midfielder earlier this week.
"No interest in Graham Dorrans," added Allardyce, after seeing his side win 2-1 against Colchester United on Tuesday.
"Graham is a midfield player and you can see how many midfielders we had out there today."
Downing is another unlikely to be signing a deal at Upton Park this summer, according to the Hammers boss.
Allardyce claimed: "I wouldn't have thought that we could afford Stewart, unfortunately."