The 18-year-old has been earning much praise in recent weeks, and attracting attention from clubs such as Juventus after being talked about as a future star.
He has represented Italy at U-18 and U-19 level despite being born in England, with his father hailing from the Mediterranean nation.
His performances at youth level have attracted Arsene Wenger, who is known for his policy of snapping up talented youngsters.
It is understood that the Frenchman has had scouts keeping an eye on the teenager ahead of a potential summer offer.
Juve and Napoli are also in the hunt, with De Girolamo highly rated in Italy despite plying his trade in League One.
The youngster has made two appearances for the Blades, and is seen as one for the future at the club.
However the third tier side are aware that a serious offer may be too good to turn down, which could see the starlet leave in the not too distant future.
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