The 25-year-old was quick to address the United fan base after a busy day training with Brazil at Tottenham Hotspur's Enfield Training Centre ahead of jetting off to the 2018 World Cup in Russian Federation.
Dalot has a £17.4m release clause and is understood to have had his medical.
Matteo Darmian has been linked with a switch to Juventus in recent weeks, which would leave Mourinho short of specialist right-backs in his squad.
United fans are unlikely to get much of a glimpse of Fred at the World Cup in Russian Federation, though, as he is yet to start an worldwide under current coach Tite.
Brazil coach Tite had said he wanted his players' domestic futures sorted before the tournament in Russian Federation, which starts on 14 June.
Fred caught the eye in Shakhtar's run to the last-16 of the Champions League, where the Ukrainians only lost out on away goals to Roma.
Lucescu is credited for transforming Fred into a holding midfielder at Shakhtar.
Brazil worldwide midfielder Fred was reportedly undergoing his medical yesterday (Monday) ahead of a transfer from Shakhtar Donetsk.
Handed his debut by Dunga in 2014, Fred's global career was disrupted after testing positive for the diuretic hydrochlorothiazide during the 2015 Copa America.
The suspension only applied to games sanctioned by South American football federation Conmebol.
Lucescu said of his imminent arrival at Old Trafford: "I'm really pleased Fred has completed his transfer to Manchester United, I am proud of him".
You can see something of a mini version of [Manchester City midfielder] Fernandinho.
United are also completing the finer details of terms with Porto for full-back Diogo Dalot, who is due in London to sign his contract, also expected to be for five years.