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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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.