Militao, 20, has a £42.5million release clause in the contract he signed last summer, and his form in the last year has helped him break into the Brazil squad.
Liverpool have sent scouts to watch the Porto star in action, while Jose Mourinho turned down the chance to sign Militao for just £3m before he signed a new deal with Porto.
The Sun claim United are now keen on bringing the Brazilian to Old Trafford, despite his price tag sky rocketing in the last six months.
It is unclear who will be in charge at United come the summer, but a deal for Militao will not be pursued until the end of the season.
Porto want Militao to put pen to paper on another long-term contract, which would see the right-backs release clause figure rise again.
Defence remains an issue for any head coach in charge at United, as the club searches for experience and solidity at the back.
Ole Gunnar Solskjaer is keen to take on the job at United permanently, but Mauricio Pochettino remains the Red Devils top choice to take charge.
Liverpool have Trent Alexander-Arnold, also aged 20, occupying their right-back slot, but Militao is being tipped to move to centre-back in the future.
Militao is under contract with Porto until 2023, but the interest of the likes of Real may turn Militaos head.
While a deal for Militao will be put on hold until the summer, United and Liverpool are not expected to do any serious transfer business in January.