Ibrox boss Steven Gerrard has proven to be an immensely popular hire after taking over from Graeme Murty in the summer of 2018.
Meanwhile, Neil Lennon has defied critics to oversee a strong start to his first full season back at Celtic.
Despite this, Sutton can't see both of them co-existing in Glasgow beyond the end of the current campaign.
He told The Herald: “If I’m honest, the team that finishes second in the league, I don’t see their manager lasting next season. I think it’s that big.
“They are both managers with great passion and knowledge of the game, but with what’s at stake, this season is massive and that starts with the first trophy, even though the Betfred Cup isn’t as big as the league and what it means to both sets of supporters.
“The Betfred Cup is always such an important competition and this season even more on the basis that the Rangers fans feel they are getting somewhere under Steven Gerrard and there’s an expectation for them to win trophies from the support.
“You look at Celtic’s domination in recent years and their fans will expect the same again."