The Red Devils are expected to be in the market for a new midfielder this summer as Michael Carrick prepares to hang up his boots.
Marouane Fellaini, meanwhile, is also being tipped to leave Old Trafford following the expiration of his contract.
According to Diario Gol, United boss Mourinho is desperate to acquire the services of Rakitic, who has made 200 appearances for his Catalan club following a move from Sevilla in the summer of 2014.
Rakitic has already made 41 appearances for Barcelona this term - scoring four times and registering four assists.
The 29-year-old, who started his professional career at Basel, has also earned 88 caps for the Croatia national team.