Cavani's existing deal expires at the end of the season, with an option for another year if both club and player agree.
Reports in South America have linked him with Boca Juniors, stating he desires to live closer to his family.
But after Cavani's excellent display against AS Roma in a 6-2 Europa League semi final first leg win on Thursday, Solskjaer hopes the striker stays.
"Edinson was out for seven months and not playing football for a very long time before he came to us," Solskjaer said.
"He's had a couple of injuries and has worked really hard to show his quality.
"He knows my feelings, he knows that I would love to have him for another year, we have spoken about it. I understand this year has been very difficult, but I promised him that Old Trafford and Manchester is a different place with fans in the stadium.
"I'm doing my best, nights like this [help], hopefully we can get to the final and hopefully he can see himself being here for another year."
Asked how he can convince Cavani, Solskjaer replied: "We don't have to sell the club to him. It's about the feeling he gets coming into training every day, the environment we created, the team we are creating that he is a part of.
"He has to feel what it's like to score those goals at the Stretford End with fans, it's a different feeling.
"We've got the best fans and, as a goalscorer, there is no better feeling than scoring and celebrating in front of them."