Rossi will have to play the waiting game to find out where he will play next season.
"Parma have made an offer and are awaiting a response from Manchester United," said the player's agent Andrea Pastorello.
Rossi joined Parma on a six-month loan deal from United and was pivotal in helping the Ducale outfit retain their Serie A status.
Although the United youngster is rated highly at Old Trafford, Sir Alex Ferguson's long-term plans for the Italian could change with the arrival of West Ham's Carlos Tevez.
The Argentina forward is expected to join United after the Copa America, a move which could see Rossi leave Old Trafford.
"I think with an offer of 10-12m euros (£6.7-8.1million), Manchester United could be tempted to let Rossi leave," added Pastorelli.