The Red Devils have endured a difficult start to the new Premier League season following three losses in their opening seven games.
Manchester United slumped to a 1-1 draw with Wolverhampton Wanderers at Old Trafford to lose ground on their title rivals last weekend, before their 3-1 loss to West Ham on Saturday.
Arsenal, on the other hand, were 2-0 winners against Everton to maintain their strong run of results and they beat Watford by the same scoreline this weekend.
Tottenham were 3-0 winners at Manchester United earlier this month before Spurs lost 2-1 at home to Liverpool FC.
Manchester United finished in second spot in the Premier League table last term to secure their best finish since Sir Alex Ferguson.
Former Arsenal striker Wright believes the Red Devils will challenge Tottenham for fourth spot.
“I think Man United are fighting for fourth depending on how Tottenham go,” Wright told Sky Sports.
“Tottenham are in a little bit of a rut. They have the quality and capability of getting in the top four.”
Manchester United finished second and without a trophy last term.
They will return to Champions League action on Tuesday night when they host Valencia at Old Trafford in the group stage.
Mourinho signed Shakhtar Donetsk star Fred, Portuguese defender Diogo Dalot and Stoke goalkeeper Lee Grant.
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