Massara and Nice sporting director Florent Ghisolfi are now being discussed inside Anfield, says the Athletic.
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Italian Massara is a free agent after being sacked by Milan at the end of last season along with fellow director Paolo Maldini.
Massara and Maldini built Milan's Scudetto winning team two years ago, but both were unceremoniously dumped by Milan owner Gerry Cardinale.