Mizuno is expected to learn this week whether he will be granted a work permit to play in Britain.
The 22-year-old JEF United player is new to international football and his shortage of caps for Japan will be a factor to be considered by the Department of Employment.
Their verdict on his suitability for a work permit could come before the end of the week.
Agent Roberto Takuda said in the Daily Express: "The interest of Celtic in Mizuno is true, but the deal is complicated.
"The player is still in Japan and if it was down to him it would be simple because he is 100% committed to making this work.
"But there are other things to be considered. There is the work permit and also talks with his club."