The Red Devils lost further ground on their title rivals last weekend as they were held to a 1-1 draw by Chelsea at Stamford Bridge on Sunday afternoon.
Arsenal, meanwhile, returned to winning ways with a 2-0 victory at Bournemouth to leave them in third place in the top flight table.
Former Liverpool star Barnes feels that the teams currently in the top four will be the same sides occupying the Champions League places come the end of the season.
And Barnes is tipping Manchester City to finish as Champions ahead of Arsenal, Tottenham and Leicester.
Speaking at a Titanbet #AskBarnes Q&A session, the former England man said: “I think the top four will remain the same as it is now.
“I think five or six games ago, I would have had Leicester drop out, but they have a massive gap on Manchester United so I think they will stay in the top four.
“Manchester City will be top and Arsenal and Tottenham will be either second or third.
“For the bottom three, I have Aston Villa, Norwich, and Sunderland. Bournemouth are doing just enough to keep them out of relegation. Man United might drop to the bottom three as well!”
Arsenal host leaders Leicester City on Sunday knowing that a win would hoist them two points behind the Foxes, while Tottenham travel to Manchester City later in the day.
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