Metro reports that both clubs had scouts to watch the 19-year-old in action over the weekend.
Moore is regarded as one of the best young defenders outside of the Premier League.
The England Under-19 star has also been linked with the likes of Tottenham, Everton and Aston Villa.
Leicester reportedly want around £5million for the youngster.
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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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