The defensive midfielder starred for Portugal at the European Under 21 Championships this summer.
And the Premier League duo are keeping tabs on the 23-year-old.
Carvalho is yet to play this season after suffering a stress fracture in July, but he is expected to return shortly in plenty of time for the January window.
Both Arsenal and Liverpool have struggled at the start of this season, with the Gunners going down against Chelsea on Saturday.
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William Carvalho is on Arsenal and Liverpool's radar
Arsene Wenger's side have ten points from six games and sit sixth in the table while Liverpool are down in 13th.
The Sporting Lisbon midfielder would cost around £25million.
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