United agreed a £37 million deal on deadline day but the 18-year-old remained with Atalanta until last week.
Diallo has now signed a four-and-a-half-year contract with United, with the option of a further season, and the winger is now waiting on a work permit.
Earlier this week, reports claimed that Solskjaer personally contacted Diallo ahead of his arrival at Old Trafford.
And the United manager believes the winger has the attributes to make a swift transition at United.
‘As a club, we have followed Amad for a number of years and having watched him myself, I believe he is one of the most exciting young prospects in the game,’ Solskjaer told United’s official website.
‘Manchester United has such a proud history of developing young players and everything is in place to enable Amad to reach his potential here.
‘It will take time for him to adapt but his speed, vision and fantastic dribbling ability will stand him in good stead to make the transition.
United have said they will confirm Diallo’s shirt number in the coming days.