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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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?