The Red Devils have endured a tough start to the new campaign under Jose Mourinho and the Portuguese coach has already found himself under intense pressure at Old Trafford.
Manchester United slipped to a third defeat of the season in the Premier League on Saturday when they were beaten by West Ham United at the London Stadium.
That defeat followed their surprise League Cup exit at the hands of Derby County last week to heap further pressure on Mourinho at Old Trafford.
Manchester United will take on Chelsea FC at Stamford Bridge on 20 October before facing Manchester City away on 11 November.
Chelsea FC have been in fine form so far this season and they are currently third in the Premier League table.
And Carragher has warned Mourinho that his team will have to seriously step their game up if they want to be ready for those tricky clashes.
Speaking to Sky Sports on Monday, Carragher said: “Manchester United have not had a tough start on paper.
“They have only played Spurs at home out of the top six.
“But when we come back from the international break, starting with Chelsea and finishing with Manchester City, that’s why you brought Jose Mourinho to your club, to win big games.”
Manchester United are currently preparing for their home Premier League clash against Newcastle United at Old Trafford on Saturday evening.
The Red Devils finished second and without a trophy last term.
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