The 27-year-old is currently 2nd choice behind youngster Hugo Lloris at the Municipal du Ray.
Dennis Wise will look at Gregorini next week before deciding whether to sign the keeper on a permanent basis.
Gregorini will be the second Nice player to train with the Whites this month.
Defender Anthar Yahia was also given the chance to win a deal with Leeds, but looks set to opt for Bochum instead.