The contract of World Cup winner Pogba, who joined United from Juventus for 89 million pounds in 2016, is set to run out in 2022.
"He knows my feelings about him and that I want him to stay," Solskjaer, whose side are second in the Premier League and in a good position to reach the Europa League final, told reporters.
"It's about creating an environment that we thrive in... and a team that challenges for trophies."
Cavani arrived at United on a one-year contract in October with an option of another 12 months but remains undecided about his future after a difficult season which included a social media faux pas that resulted in a three-game ban.
"He (Pogba) and Edinson are winners but they're also human and it's important that they enjoy coming in and enjoy playing for this team," Solskjaer said.
"If... they can feel our progress, our chances are probably bigger to keep them."
Pogba's agent said in December that the player was unhappy and needed to change teams to reignite his career.
But Solskjaer said his relationship with Pogba, who he also coached at youth level, had helped the Frenchman rediscover his best form.
"He wants to be better, he has listened, and we can see that he's enjoying himself here as well," Solskjaer said.
"I had him when he was a kid as well and I've always said I enjoy managing Paul; I enjoy coaching him, talking to him and challenging him because he's a winner."