The Red Devils are said to be prioritising the signing of a new midfielder at the end of the season after strengthening in attack during the most recent window.
With Michael Carrick also due to depart Old Trafford in the summer, Mourinho has now started his search for a reliable performer to help bridge the gap on Manchester City.
Kroos is seen as the ideal candidate to slot in alongside Nemanja Matic and Paul Pogba in a three-man midfield, and The Independent suggests that United are confident of landing their main target following previous near misses.
Madrid are themselves looking to bring in fresh faces, it is claimed, and the Germany international is among those who will be available to purchase at the right price to make room.
Man City have also previously been linked with Kroos, however, and may rival United in making an offer in the summer.