La Nazione says Newcastle are considering a club-record £53.7million move for La Viola's No25.
The move, however, is likely to hinge on whether the Saudi Arabia Public Investment Fund's controversial £300m takeover from Mike Ashley is ratified by the Premier League.
The Magpies' current record-buy is two-goal striker Joelinton - a £40m buy from Hoffenheim last summer.
As for United, club bosses may see Chiesa as a cheaper alternative to No1 target Jadon Sancho.
Manager Ole Gunnar Solskjaer is believed to be keen on recruiting a new winger, striker and centre-back this summer.