Reports claimed Pogba failed to make the trip to Cardiff in an attempt to force a move to Real Madrid.
Solskjaer also confirmed the £80m signing of Leicester centre-back Harry Maguire is set to be announced “very soon”.
Penalties win in final pre-season game
In addition to Pogba, Romelu Lukaku was also missing as United drew 2-2 at the Principality Stadium on Saturday.
Lukaku’s absence throughout pre-season has also been attributed to injury, but the Belgium striker’s future has been the subject of speculation all summer, with Inter Milan and Juventus reportedly interested.
United led through Marcus Rashford, before former Liverpool midfielder Suso and Samu Castillejo put Milan in front.
Jesse Lingard levelled up to take the game to spot-kicks.
And Welshman Daniel James scored the winning penalty in the Principality Stadium.
Lukaku continues to be linked with a move away from Old Trafford.
United start their Premier League campaign at home against Chelsea on Sunday, 11 August.
Victory for United against Milan means they have won all six of their pre-season matches, albeit with this one requiring a penalty shootout, which finished 5-4.
Daniel Maldini, the 17-year-old son of Italy and Milan legend Paolo, missed the only spot-kick.
The result means Portuguese side Benfica win the 2019 ICC competition, with United having needed to win by three goals or more to claim the trophy.