Mourinho is searching for a long-term replacement for Michael Carrick, whose contract is set to expire at the end of the season.
Mourinho said recently: "If - next summer - we are going to sign a midfield player, it's to replace Michael Carrick".
Brazil global Fred has scored two goals in 16 league games for Shakhtar Donetsk so far this season and has also featured six times in the Champions League.
And Mourinho knows he needs at least two players with Carrick likely to leave Old Trafford.
"And Michael is a phenomenal player that, this season he couldn't give us anything at all".
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Fred, 24, is part of Mourinho's plan to bolster his squad with two new midfielders.
Fred is believed to have been targeted by the Portuguese manager, but is only one of several other players being monitored.
It is understood that Mourinho is also chasing Valencia's Carlos Soler and Atletico Madrid's Saul Niguez, but the latter's release clause of 140 million might deter further advances.
Fred stands at Shakhtar since July 2013.