Rangers will take on Old Firm rivals Celtic in the semi-finals of the prestigious cup competition next month.
Rangers fans will believe that they can actually get the better of Celtic in the semi-finals of the Scottish Cup.
Former Celtic striker Sutton has suggested that Gers boss Warburton could be under pressure if his side get thumped by the Hoops.
The former Chelsea man has also revealed that the former Brentford manager will have to make Rangers competitive next season, as fans will not settle for anything less than a title challenge.
“What happens if Rangers go to Hampden and get thumped 4-0 or 5-0 by an average Celtic team?” Sutton wrote in The Daily Record. “I would suggest that puts Warburton right back to square one."
“You talk all you like about settling-in periods and Rangers being given time by their fans to find their feet in the top-flight again but you and I know that is absolute rubbish," added the Englishman.
“Are you telling me Rangers fans will be happy with digging in for a point at Inverness to secure a place in the top six next season while Celtic and Aberdeen are 30 points ahead fighting for the championship title? Of course not.
“That position would be real pressure for Warburton who seems to get irritated at the slightest thing. And who knows for sure if he’d fancy it?”
Rangers are at the top of the Scottish Championship table at the moment with 69 points from 27 matches. The Gers are 12 points clear of second-placed Falkirk and have a game in hand.