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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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.