Milne, 30, was signed two years ago on a free transfer from the Scottish Premier League side Falkirk but Iron fans have rarely seen him on the pitch.
The Scotsman managed just two games in August 2008 before knee problems kept him out of action for 17 months.
He returned at the end of last season before suffering a further setback. He attempted to return during United's first game of pre-season at Brigg Town last week, but lasted just half an hour.
Adkins fears that Milne will now not be able to add to his total of eight competitive appearances for the Iron.
"We've got to make a decision and Kenny has to as well at some stage about just how far we can keep progressing. you feel for the big man because he's been a great, great professional," he told the club's official website.
"We will have discussions during the course of the week. He's been fantastic trying to get himself right, so we've come to a scenario now where have we actually achieved it or not.
"It's so difficult because Kenny is a great professional, great person, and a good player Whatever way you want to talk about it, it's his career, and he's not in a nice place. We've got to help him as much as possible, as we have done, but there has to come to a stage where decisions are made."