Hutchinson has reportedly agreed a pre-contract deal which will see him arrive at Celtic Park in the summer.
However Southgate was not forthcoming with any confirmation of the move.
The 20-year-old has made several appearances for the Middlesbrough first team but was not required for Saturday's FA Cup clash with Mansfield.
Alerted to reports concerning Hutchinson, Southgate said: "That's something that has emerged over the last couple of days, and I haven't even had a chance to sit and speak to Ben about it.
"I don't want to say too much about that at this moment in time because there is a bit of investigating to be done into it."