The Manchester Evening News claim that Pep Guardiola was pursuing a deal for the 25-year-old in January but pulled out at the eleventh hour because he did not feel that there was any rush.
Guardiola met with City's CEO Ferran Soriano and sporting director Txiki Begiristain in December and decided that they needed cover and competition for Fernandinho, another Brazilian signed from Shakhtar in 2013.
City subsequently identified Fred as the ideal challenger to Fernandinho, who at the age of 33 may not have many years at the top of the game left in him.
But as Fernandinho's strong performances continued into the second half of the season, Guardiola decided that there was no need to rock the boat by signing Fred at that time.
It would also have been a difficult transfer to complete, as Shakhtar were still in the Champions League at the time and would have demanded a hefty price for one of their most prized assets.
The Ukrainian club wanted £50m for Fred and were keen to get United involved in a bidding war to drive up the price even higher. It seems that they may yet get their wish, with Fred quoted by the BBC as saying that both Manchester clubs are interested in his signature.
United are believed to be in the driving seat with City now prioritising the acquisition of Jorginho from Napoli instead. United would have preferred to sign Sergej Milinkovic-Savic from Lazio but I Biancocelesti's demands were too high.