RAI Sport says Matuidi's agent Mino Raiola is keen to engineer a move out of Juve for the France international this summer.
And he has offered the midfielder to United boss Jose Mourinho.
Matuidi only joined Juve last summer from PSG and has stated he is enjoying his football in Turin.
However, Raiola has spoken to Mourinho about Matuidi's situation and received encouragement from the United manager. The Portuguese has stated in recent weeks he is targeting two senior midfield signings this summer.