The 23-year-old has been nursing a foot complaint in recent months, the same problem which saw him withdraw with England duty in March.
However, Rashford has been passed fit for Thursday’s first leg at Old Trafford. Solskjaer said: “We’ve managed him and we’ve had to manage him for quite a while actually.
“Of course last year with his back, then he had his shoulders, which he’s recovered really well from, and I feel he’s gone through the worst of this foot (issue) as well and he almost played a full game against Leeds.
“He’s available for selection again on Thursday, so I think he’s happy with the progress that he’s made.
“We didn’t lose any anyone after the Leeds game, which was a great preparation for us because it’s a high intensity, very, very difficult game.”
Solskjaer kept schtum on who will be in goal for United on Thursday, although David De Gea now appears to have taken over as cup goalkeeper with Dean Henderson between the sticks for Premier League matches.
Manchester United predicted XI (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Fred, McTominay; Pogba, Fernandes, Rashford; Cavani
Manchester United squad from: David de Gea, Lee Grant, Dean Henderson, Nathan Bishop, Victor Lindelof, Eric Bailly, Phil Jones, Harry Maguire, Luke Shaw, Alex Telles, Aaron Wan-Bissaka, Brandon Williams, Axel Tuanzebe, Will Fish, Paul Pogba, Juan Mata, Fred, Bruno Fernandes, Amad Diallo, Daniel James, Nemanja Matic, Donny van de Beek, Scott McTominay, Ethan Galbraith, Shola Shoretire, Edinson Cavani, Anthony Martial, Marcus Rashford, Mason Greenwood, Anthony Elanga