The former Arsenal defender is attracting major interest from the Premier League and also within Serie A.
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Italian giants Juventus and Napoli are both interested in Norton-Cuffy, while a return to England is also on the table.
TalkSPORT says Newcastle, Everton and West Ham are all watching the fullback this season.
Norton-Cuffy left Arsenal for Genoa two years ago and is now regarded as a first-choice for coach Daniele de Rossi.