The 26-year-old striker, who is currently out of contract, has spent time on trial with the Championship club this summer.
Although the Stags have offered Green a new deal, Cox believes that he will leave Field Mill.
"Judging by how he has been talking, he is likely to sign for Birmingham. We have been planning for life without Matt," Cox told BBC Radio Nottingham.
"Birmingham have put in an offer in front of him and they are still talking. I am not sure if Matt has accepted it or not. Matt wants to play in the Championship. We made a very good offer, but I think it is more down to ambition rather than the financial aspect.
"He has [also] spoken to a number of League One clubs, but he said if he doesn't sign for Birmingham he will come back here."
Green scored 25 goals last season as Mansfield secured promotion to the Football League.