Miyaichi has made just seven appearances for Wigan this season due to injury and has not started a game for the club in the Premier League.
The 20-year-old is now likely to miss the rest of this campaign after undergoing surgery on an ankle injury suffered in last weekend's FA Cup win at Everton.
That was his first appearance for Wigan since November, but Martinez is keen to work with Miyaichi again next term if Arsenal agree to another loan.
"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 Wigan 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 after being 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."