La Gazzetta dello Sport reported the Red Devils will contest for the 20-year-old's signature alongside the likes of Juventus, Inter Milan, Liverpool and Premier League champions Manchester City (h/t Football Italia).
It's said Napoli have already offered €50 million (£43.7 million) plus midfielder Marko Rog, and it's expected the level of interest in his services will take that figure higher.
Italian journalist Tancredi Palmeri provided more context, adding that AS Roma are also in the hunt:
United brought Alexis Sanchez to the club from Arsenal in January, further complicating first-team aspirations for Anthony Martial and Marcus Rashford, who have struggled for game time.
Jose Mourinho has a reputation for preferring tried-and-tested senior talents ahead of up-and-coming assets, posing the question of whether he'd be the manager to squeeze the most out of Chiesa.
United will also need to move efficiently in the market if they're to snag a deal for one of Serie A's most highly rated prospects, as Italian writer Vince van Genechten pointed to the interest of other big outfits:
Chiesa has already been capped four times by Italy and looks set to play a large part in the Azzurri's remodelling under Roberto Mancini.
The youngster started in the lineup that lost 3-1 to France in a friendly meeting on Friday, but the Evening Standard's Tom Collomosse nevertheless picked the emerging star out for particular praise:
Fiorentina have on their hands a wide technician who is capable on either flank but could risk his development charting off its current course if he moved abroad, a risk United are likely to be well aware of.
With some of Italy's juggernauts already in the race and a raft of clubs seemingly interested, United face a fight if they're to beat their widespread competition and add a rare Italian recruit to their ranks.