BBC Sport says Liverpool have offered €30m for the teen - with Parma ready to accept.
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Leoni enjoyed a breakthrough year in Serie A last season, helping Parma beat the drop under the guidance of former coach Cristian Chivu.
Leoni has been linked following Chivu to Inter Milan this summer. However, he is now on the brink of moving to Liverpool.
Taking on 18 year-old Leoni will not see Liverpool end their pursuit of Crystal Palace defender Marc Guehi.