The 26-year-old, who started in France's 3-1 victory over Senegal in the World Cup on Tuesday, is seen as an option to bolster Michael Carrick's engine room.
United have already agreed to sign Atalanta midfielder Ederson for £35m on a four-year contract, and Tchouameni could soon follow him through the doors at Old Trafford.
Real Madrid are reportedly open to selling the Frenchman for around €80m (£69m), and Cadena COPE commentator Ruben Martin believes offloading the combative midfielder could open the door for Jose Mourinho to Mateus Fernandes.
"What I'm saying is that if Real Madrid keeps Tchouameni and Manchester United signs Mateus Fernandes, the player Mourinho really wanted was Mateus Fernandes.
"So you're closing the door on yourself if you don't sell Tchouameni to United. If you sell him, you'll recoup some money, and you can sign the player Mourinho wants," Martin said.
Fernandes is also on United's radar, but they are unwilling to meet West Ham's £80m asking price. In this scenario, Real could sell Tchouameni to the Red Devils, and Mourinho can sign one of his top targets.