The Sun says the decision has been made at board level, although the club may not announce it until the end of the season.
United are publicly sticking to the mantra that the search for Jose Mourinho's permanent successor remains ongoing.
But the hunt is over with co-chairmen Joel and Avram Glazer now convinced they have the right man in Solskjaer.
Avram Glazer travelled from America to be at United's 3-0 victory over Fulham at Craven Cottage on Saturday.
He spoke with Solskjaer after the game in the dressing-room, having already talked to the interim chief's agent Jim Solbakken, who had sat with the directors as a guest of United.