The England striker suffered an ankle injury during the goalless draw with Liverpool last Sunday and was forced to watch from the sidelines for United's midweek win at Crystal Palace.
Fellow forward Anthony Martial has also been nursing a knock and Solskjaer says the Frenchman has an outside chance of featuring against the Saints at Old Trafford.
Meanwhile, Solskjaer has revealed Antonio Valencia, who will see his contract run out this summer, is not fully fit and the Norwegian coach does not know what the future holds for the veteran.
Solskjaer said: "Antonio is still not 100% fit. He has had a fantastic career here at Manchester United and in England, and is one of the Premier League winners left in the dressing room.
"But at the moment I'm not sure if Man United and Antonio will agree on the next year. Hopefully I can get him on the pitch and he can show what he can do."