United have been heavily linked with a deal to sign the French international in the last 18 months but no firm offer has materialised from the La Liga giants.
Pogba has previously hinted at his desire to move to the Spanish capital but United have fought hard to keep hold of their former academy player.
According to reports from Sky Sports, via Diario AS, Pogba plans to see out the final year of his contract at Old Trafford in 2022 and join Real Madrid as a free agent.
United are rumoured to be willing to let him leave next summer after admitting defeat in their contract extension talks with the 28-year-old World Cup winner.
Ole Gunnar Solskjaer wants to retain his midfield lynchpin for the upcoming campaign with the Red Devils pushing to launch a Premier League title bid in 2021.