The 29-year-old has recently been linked with a move to Arsenal, while a host of other Premier League clubs are also thought to be interested in his services.
However, Solskjaer has revealed that he will do all that he can to convince the Scotland international to remain at the Cardiff City Stadium next season.
"David was excellent for us last season, the best goalkeeper in the Premier League," he told Wales Online.
"I am hoping to convince him to stick with us. It's up to me really to sell the idea of what we are going to be trying to do here next season and beyond that, sell my philosophy to him if you like.
"I know he likes it at the club and is settled in the area, so we are hopeful he will be with us next season."
Marshall has four years left to run on his contract with Cardiff.