The Times reports that several Premier League clubs are hoping to sign the 20-year-old on a loan deal in January.
Zaha is considered one of England’s top football talents and joined United from Crystal Palace over the summer.
The youngster featured for United in the Community Shield win over Wigan in August but has not played since.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?