Cavani, 34, was at his mercurial best as the Red Devils romped to a 6-2 victory over Roma in the first leg of their UEFA Europa League semi-final, scoring twice and setting up another two goals for teammates Bruno Fernandes and Mason Greenwood.
The veteran striker's match-winning performance comes amid speculation linking him with a move away from Old Trafford, with recent reports suggesting that he has opened talks with Argentine giants Boca Juniors regarding a potential return to South America.
United boss Solskjaer is aware the Uruguay international is considering his future, but revealed he is hoping the former Napoli and Paris Saint-Germain star will have a change of heart after hitting top form for the club.
"He was out for seven months and he'd not been playing football for a long, long time before he came to us," the Norwegian said, per Manchester Evening News.
"He's had his couple of injuries and he's now worked really hard to get back fit and he shows his qualities.
"He knows my feelings. He knows that I'd love to him for another year. We have spoken about it. I understand this year has been very difficult but I've promised him that Old Trafford and Manchester is a different place with our fans in the stadium.
"He must try to get that feeling, to be on our side. Manchester is a fantastic place to live. I'm doing my best so let's see. Of course, after nights like this, hopefully, we can get to the final and hopefully he can see himself being here for another year."
