Benfica's Swedish international Victor Lindelof has become Jose's top January transfer target.
United are believed to have made contact with the Portuguese champions about the 22-year-old centre-back ahead of a proposed £38 million move.
Phil Jones and Marcos Rojo have done a magnificent job at the back for Jose's patched up defence in recent weeks, but Jones has a history of lengthy injuries and that cannot be ignored.
Chris Smalling came on late at West Brom and is working his way back to full fitness but Eric Bailly will be unavailable in January.
The Ivorian will be away with his country - the defending champions - at the African Cup of Nations, and should they reach the final again - as expected - he could be in Gabon for up to five weeks.
With a dearth of defenders at his disposal, Jose wants to strengthen his options at the back and has touted Lindelof as the man to do so.
Despite the fact Benfica are still in the Champions League and could be unwilling to sell, their president Filipe Vieira has met United officials in an attempt to thrash out a deal to bring the centre-back to Manchester.
Mourinho has watched Lindelof on several occasions and is known to have been impressed by the Swede's physical, bruising and no-nonsense style.
Lindelof has made eleven senior appearances for Sweden and started all three of their group games at last summer's European Championships.
Sweden exited at the group stage but a first goal for his country soon followed for Lindelof when he netted against Bulgaria in a 3-0 World Cup Qualifying win.
He started his career with his hometown team, SK Vasteras, in Sweden, but moved to Benfica in 2012.
Lindelof made 97 appearances for their B side and then made a senior bow the following year, going on to play more than 50 times for the Estadio da Luz outfit.
He was part of the victorious Sweden squad at the 2015 Under-21 European Championship and scored the winning penalty in the shootout against Portugal in the final.
Mourinho is also set to clear out some of his squad's dead wood in January.
Bastian Schweinsteiger, Morgan Schneiderlin and Memphis Depay all look likely to leave, with Everton heavily linked to the latter two.
Much-maligned midfielder Marouane Fellaini has also emerged as an unlikely transfer target for both Milan clubs.