The Japanese player has made just seven appearances for the Latics in an injury-dogged season which could have been ended by an ankle injury in last weekend's FA Cup quarter-final win at Everton.
Miyaichi requires surgery to repair the ligament damage and it may rule him out of Wigan's remaining games.
But Martinez is a huge fan of the 20-year-old and would be keen to extend their working relationship if Gunners boss Arsene Wenger will allow it.
"We will have to speak with Arsenal and we need to assess and make sure they are happy with the progression of the player," said the Latics boss.
"Ryo is a footballer that we really admire and has an incredible future in front of him so if we could work with him for a longer period we would be very happy to do that."
Wigan have had their fair share of injuries this season, critically to players like centre-back Antolin Alcaraz who has made just six appearances, having been sidelined for nearly six months with a groin problem.
Martinez believes he has noticed an upward trend in injuries among all clubs and has an interesting theory as to a contributing factor.
"Don't get me wrong, football in general gets more demanding physically, that is clearly the way the game is developing," he said.
"The more technology in sports science means you create better athletes but I think the weather and rain we had this summer has created difficult surfaces to play and train on.
"It would be interesting to find out how many injuries clubs throughout the leagues have had compared to last year because the number has been much higher.
"I don't think the physicality of the league has anything to do with that."
Martinez, preparing his side for the visit of Newcastle tomorrow, believes this year's relegation fight is the toughest he has experienced.
"The teams in the bottom three I don't think are going to stay there: there will be other teams involved and it will go down to the final day once more," said the Latics boss, whose side dropped into the relegation zone because of their FA Cup involvement last weekend.
"Year after year that fight becomes harder because teams are more prepared and stronger to cope with the demands in that final push.
"I do see it as the strongest fight out of the last four I've been involved."