The Sun says the Italy midfielder, 22, was among the targets identified but not followed through during United's difficult summer transfer window.
Pellegrini fits the profile of a box-to-box midfielder to give the team a more dynamic link between attack and defence but a new centre-back remains manager Jose Mourinho's priority.
Whether United go back for Tottenham's Toby Alderweireld or another centre-half will indicate the balance of power between Mourinho and his Old Trafford bosses.
There is doubt inside the dressing room over whether the Special One will still be in his job by the January window.
Mourinho has told friends he is frustrated and his performance at Friday's press conference showed the stress he has privately described.