Pogba's deal expires at the end of the current season and he is yet to agree to fresh terms, giving way to speculation that he could leave Old Trafford at the end of the current campaign.
If he does not sign a new contract by the turn of the year, Pogba will be free to discuss terms and sign a pre-contract with a new club during the January transfer window.
According to ABC, Real are ready to offer him just under £12m-a-year with a transfer bonus of just over £25m.
The report also claims that Pogba's agent, Mino Raiola, is pushing for the deal to go through, and Madrid are hoping to pair Pogba with his French teammate, Karim Benzema.
Earlier this week, AS reported that Los Blancos believe signing Pogba will facilitate them to make a move for Mbappe, who is their number one target. However, Mundo Deportivo has claimed that Carlo Ancelotti's side are unwilling to enter a bidding war for Pogba with other potential suitors.