The move comes at the request of the Naples club's manager, Antonio Conte.
United have already done business with the Azzurri, who signed Scott McTominay last summer.
According to Il Mattino, Zirkzee may be on the verge of leaving United this winter.
He has struggled to make a significant impact on the team after arriving in the summer.
Many fans are unsure how he will fit into new manager Ruben Amorim's 3-4-2-1 system.
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