Green, who had a trial with Blues this summer, is out of contract after helping Mansfield secure promotion back to the Football League last season.
A "very good offer" is on the table from the Stags in an attempt to keep the 26-year-old at the One Call Stadium, but Cox believes Green is bound for the Blues.
"I understand Matt's desire to play at the highest level possible and I'll be respectful of whatever he decides, but I've got to do what is best for Mansfield Town," Cox told the club's official website.
"We've made Matt a very good offer and he has stated that if he isn't going to sign for Birmingham, he wants to come back here. But from what I can gather, Birmingham are still very keen to sign him."
Meanwhile, Cox confirmed that goalkeeper Ian Deakin, who played for non-league Buxton last season, is training with the squad.