Miyaichi was hurt after coming off the bench during Saturday's FA Cup quarter-final win at Everton.
His scan results are not yet back, but Martinez is planning for Wembley and the relegation run-in without the 20-year-old.
Japan international Miyaichi, who joined on a season-long loan last summer, spent three months out around the turn of the year with a different ankle problem.
Wigan's Twitter feed carried Martinez's latest bulletin and also said an injury suffered by Callum McManaman at Goodison Park was not serious.
You can take the boy out of Arsenal, but you can't take Arsenal out of the boy....- bergkamp