The Red Devils have been short of quality in that department this season, with Daley Blind often having to fill in as a makeshift central defender while the likes of Chris Smalling and Phil Jones have struggled with both fitness and form.
Latest reports claim United are leading the chase to sign in-form Burnley star Keane, who has also been linked with rivals Chelsea, Liverpool and Everton in recent times.
However, Mourinho also has one eye on one of his former players as he weighs up a reunion with Real Madrid defender Varane at Old Trafford, according to the Daily Telegraph.
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Manchester United fixtures
Swansea (H) – Sun 30 April
Celta Vigo (A) – Thu 4 May
Arsenal (A) – Sun 7 May
Celta Vigo (H) – Thu 11 May
Tottenham (A) – Sun 14 May
Southampton (A) – Wed 17 May
Crystal Palace (H) – Sun 21 May
The France international made his breakthrough at the Bernabeu during Mourinho’s time in charge, and has long been regarded as one of the most promising young players in Europe in his position.
Despite signing Eric Bailly from Villarreal last summer, it may be that Mourinho has Keane and Varane in mind as his first choice central defensive partnership for next season.