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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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.