Byrne was released by Alan Knill last summer, but Brian Laws is eager to bring him back to Glanford Park.
“Contact has been made and Scunthorpe have expressed a desire to bring Cliff in,” said Johnson.
“It has to be right for both parties but most importantly of all, for Cliff himself.
“At the moment, the situation is no clearer. I would like to keep him here. We get on great and though I haven’t seen the best of him football-wise due to injury, he has been excellent behind the scenes.
“I know that he will end up being a manager one day. He is a real, true leader of the sort that is very hard to come by.
“It is difficult at the moment. We want to do the best by Cliff.”
Speaking to the Scunthrope Telegraph, Laws, who is also keen on re-signing Byrne’s former United team-mate Andy Dawson, said he had planned to ‘investigate the potential’ of being able to bring Byrne back.
"I think it was a huge mistake to let him go.
“I will see if there is a possibility, but the one thing that would be a stumbling block is that he’s got that extra year at Oldham".