The 23-year-old, who was recalled from Fiorentina earlier this month, is free to head out again after Wolves sealed the signing of Willian Jose from Real Sociedad.
Since making an unlikely return from La Viola, Cutrone has made four appearances – the most recent being a start at non-league Chorley in the FA Cup last Friday – in gold and black without scoring.
Udinese, 14th in Serie A, are keen to land him for the remainder of the season.
Cutrone joined Wolves in the summer of 2019 from AC Milan for £16million but struggled to settle in England.
In total, he has made 28 appearances and scored three goals.