It's suggested the move would cost in excess of £80m.
The transfer request comes as Mourinho prepares to be formally announced as Real Madrid head coach following Florentino Pérez's re-election as president on Sunday. The Portuguese manager signed a three-year deal with the club, with Real Madrid paying €15m to Benfica — where Mourinho had been — to secure his release.
Fernandes, a box-to-box midfielder known for his work rate, ball recovery and driving runs through the middle third, scored five goals and provided five assists in 42 Premier League appearances this season. West Ham United are under significant pressure to generate transfer income following their relegation to the Championship, with their asking price set at approximately £80m.
Paris Saint-Germain and Manchester United are also understood to be interested in Fernandes, who joined West Ham from Southampton last summer for over €42m. He has made one senior Portugal appearance and was not selected for this summer's World Cup squad.
Whether Real Madrid's pursuit becomes a formal bid will be one of the first tests of the Mourinho era.