The 25-year-old has been a long-term target for the Red Devils and it was recently reported that he is the preferred option to replace Andre Onana next season.
However, they look set to miss out on his services to City. Costa has given the green light for a transfer and City are poised to enter negotiations soon.
Pep Guardiola are prepared to pay his £64 million release clause. They could recoup part of the funds with the departure of Ederson to Saudi Arabia.
The news is a huge setback for the Red Devils. The club may have to find an alternative option to replace Onana between the sticks next season.
Espanyol's Joan Garcia has been closely monitored by United, but Arsenal appear in pole position to land his signature at the end of the campaign.