The Japanese winger is a fringe player at the Emirates, and the Eredivisie champions are eager to bring him to Amsterdam after missing out on Luciano Narsingh
The Japan international spent the second half of last term in a loan stint at Bolton, with opportunities in Arsene Wenger's first team limited.
Ajax coach Frank de Boer is thought to be in the market for a new wide-man and missed out on Luciano Narsingh, who recently moved to rivals PSV from Heerenveen.
With Miyaichi having acquired the experience of playing in the Eredivise after a temporary spell at Feyenoord in 2011, the Amsterdam giant is weighing up a move for the 19-year-old Japanese.
The decision is believed to be in the hands of Wenger, who will choose whether to keep him in north London or send him to Netherlands to earn more playing time in the first team.