Pellegri was former Red Devils manager, Jose Mourinho’s transfer target during his time at Old Trafford.
However, Solskjaer now wants to bring the 19-year-old striker to Old Trafford this summer in a bid to strengthen his squad ahead of next season.
Pellegri joined Monaco from Italian club, Genoa in January 2018.
The Italian youngster has struggled with several injuries at the French Ligue 1 club in the past two years.
His current contract with Monaco will expire in 2022.