The Portuguese winger pulled up after half-an-hour of Wednesday's reserve match between Leeds and Grimsby Town.
Dennis Wise, who has not received any expressions of interest in the former Benfica and Ionikos player, hopes a suitable loan move will help Da Costa adapt to English football.
Wise said: "That's killed it. He's pulled his hamstring. He needs games, and a lot of them. That's why I want to send him out on loan and get him used to this environment."