The Red Devils, who are in fourth place in the Premier League table, are three points ahead of Spurs in the race for a Champions League spot.
Louis van Gaal’s side have won just three of their last five games, whilst Arsenal were 2-1 winners at Old Trafford in the FA Cup quarter-final on Monday night.
Former United captain Robson admitted his concern that the Red Devils could see their fourth-place challenge unravel if Spurs get three points on Sunday.
Robson posted on Twitter: “Must win game against Spurs or top 4 finish could end @ManUtd #mufc”
United are two points ahead of their nearest rivals Liverpool ahead of the visit of Mauricio Pochettino’s side.
The League Cup runners-up can move level on points with the 20-time English champions if Spurs beat Louis van Gaal’s side.
Spurs were 2-1 winners at Theatre of Dreams last season thanks to goals from midfielder Christian Eriksen and former City striker Emmanuel Adebayor.
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