The central midfielder, capped 42 times by South Korea, has returned to the Far East after a successful trial at The Hawthorns.
Albion officials met the players' representatives on Tuesday morning about a possible deal with the Championship promotion chasers.
Mowbray told the Birmingham Mail: "We still have to make a decision. We've seen him train but because we've had such and intense run of games there has been hardly any training and the players have just been ticking over.
"He's got undoubted quality as you would expect from someone who has played 40-odd games for his country.
"He plays for the biggest club in South Korea and is a big name out there and we will have more discussions today although he's flown back because he's getting married.
"He's a very creative player and has got very talented feet but I also need to consider that he would be competing with Teixeira, Gera, Koren, Morrison and Green - even so he is a big talent. But it will only happen if it suits our club.
"We won't be putting a great amount of funds into what would be a gamble."