Cole believes United's captain has done enough for the club and needs to move to a more successful team.
"If I’m Bruno Fernandes, I’d absolutely be looking for a move away from Manchester United," Cole told Paddy Power. "He’s the first butterfly and he’s getting all the arrows from everyone.
"He must go home at night and wonder why everyone is digging him out when he’s the only one pulling his weight in this team. There are others doing okay but he is literally dragging that club through this dire moment.
"When all is said and done and Bruno leaves United and maybe they return to the top, they’ll look back at this era and the only shining light will be Bruno Fernandes.
"He’s a fantastic footballer and you could put him into most teams, and he’d thrive, but I don’t think he’s won a league in his career and he needs to go and prove to himself that he can play in a top side because he’s run his race at Manchester United.
"He should leave that club with everyone; the owners, fans and the manager giving him his flowers because he gets all the stick. And that’s part and parcel of being a Manchester United footballer but how must he feel when he’s getting all the pelters despite being the one with the goals, assists and driving the team forwards.
"He’s a fabulous player and if I was him, I’d be looking to go and play in a team capable of winning trophies."