The 19-year-old joined the club from the Gunners in the summer, but picked up a groin injury on international duty with Japan and has not played since October.
"We'll need to see how he responds to the work this week as the intensity of his preparations go up a level before we decide whether he can make the trip to Liverpool or not," Martinez told the club's official website.
"But it's great news that he is back training with the squad. The injury was a big blow to him after a bright start to the season, but he remained positive throughout his recovery and is very close to 100% again, which is great news for the team."
Miyaichi impressed on loan at Bolton Wanderers last season.