Rashford has not completed 90 minutes for United since the win over West Ham in mid-March having injured his ankle in the previous weekend's Manchester derby, starting but coming off as a substitute in six games since.
The 23-year-old managed 85 minutes of Sunday's goalless draw against Leeds at Elland Road - his longest spell on the pitch for a month - and Solskjaer believes he is now close to a full recovery.
"Marcus, we've managed him. We've had to manage him for quite a while actually. Last year with his back, then he had his shoulder which he's recovered really well from. I feel he's gone through the worst of his foot as well," Solskjaer said.
"He almost played a full game against Leeds, he's available for selection again tomorrow but I think he's happy with the progress that he's made."
"We didn't lose anyone after the Leeds game which was great preparation for us because it was a high intensity and very difficult game," the United manager added.
The two-legged tie with Roma will be Solskjaer's fifth semi-final in charge of United, having lost all of the previous four.
The Norwegian said he has looked back on those previous disappointments in the hope of finally clearing the hurdle this time, but he also feels that his side are in a better place than on those four other occasions.
"We've looked at all the semis and the reasons why but as Luke [Shaw] said, every semi-final is a tough one. You play against very, very good teams. Some defeats you dwell on a little bit more, you analyse a little bit more," Solskjaer said.
"It's about putting things right but it's arriving in this moment with an almost fully fit squad. We're in a better position.
"Of course Anthony is injured but we've had situations where maybe we've rested a couple that we thought we needed for a game the next week which has been criticised and talked about but we made those decisions with the right reasons behind it.
"Now we're going to go all out. We've got Liverpool in between but that will take care of itself."
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