The Daily Mail reports that Moyes is keen to bring the Belgian forward to Goodison Park in January in a loan deal.
The 18-year-old joined Chelsea in a £18million deal from Anderlecht in the summer, but has failed to become a regular at Stamford Bridge.
Chelsea could decide to keep Lukaku at the club in January, should Nicolas Anelka or Didier Drogba leave.
Drogba has been linked with moves to Los Angeles Galaxy and Russian side Anzhi, while Anelka has been strongly linked with a switch to Chinese Super League side Shanghai Shenhua.
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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.
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