The Japan international was injured after coming off the bench during last weekend's FA Cup quarter-final victory over Everton.
The report claims Miyaichi's scan results have not been revealed yet, but Martinez is not confident the pacey winger will return to play a part in Wigan's fight for survival.
Miyaichi joined Wigan on a season-long loan deal last summer, but has struggled to stay fit in the north-west. He spent three months on the sidelines in the New Year with a separate ankle problem.
The 20-year-old is widely regarded to be one of the quickest players in the Premier League, and has spent time on loan at Feyenoord and Bolton Wanderers in the past two seasons.
The Japanese winger replaced Callum McManaman during the FA Cup tie, but picked up an ankle problem after a challenge by Everton forward Kevin Mirallas. Miyaichi was then stretchered off the field.
Wigan went on to win the quarter-final 3-0 at Goodison Park, booking their place in the semi-final against the winner of the Blackburn versus Millwall replay.
But Roberto Martinez's side still face a battle to retain their Premier League status. The Latics currently sit 18th in the table, three points away from safety.
The news of Miyaichi's injury comes so soon after Jack Wilshere was ruled out for three weeks with an ankle injury.
Arsenal face a crunch fixture against Bayern Munich in the Champions League on Wednesday, but the Gunners will have to do without Wilshere for the trip to Germany.