McCoist delivered the Third Division title last term and is under pressure to win the new Scottish League 1 this time.
But former chief executive Green - who is to return to Rangers as a consultant - believes the Light Blues boss must also win one of the cups.
Speaking as Gers prepare to take on Forfar in the Scottish League Cup, Green told the Scottish Sun: "This year is a different year. He has had a pre-season and he has been able to sign players.
"Ally has got to win the league - and got to win a cup. That's the stark reality of life.
"Ally is a fantastic guy. He is brilliant with the players, fans and media.
"But first and foremost, a manager has to win matches and has to win competitions.
"He knows he has to win the league and, in my opinion, has to win a cup as well. If he doesn't do that, he has a problem."
Green's return was announced hours before it was revealed that a group of shareholders want the removal of key board members, including new CEO Craig Mather.
Green added: "What is happening now is not what Rangers need.
"Rangers need stability. If I can help with the stability, I will."