Marshall has been linked with a host of clubs following his consistently impressive performances during the 2013-14 Premier League season.
Although Cardiff finished bottom of the table and are now gearing up for a return to the Championship, the former Celtic and Norwich City keeper caught the eye throughout the campaign.
And Solskjaer believes Marshall can be persuaded to remain in south Wales.
The Norwegian told Wales Online: "David was excellent for us last season, the best goalkeeper in the Premier League.
"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."
Although interest in Marshall has been widely reported, Solskjaer insists Cardiff have yet to receive a substantial offer.
"There have been no big bids for David yet and to be honest I don't want any big bids for him either," added the former Manchester United forward.
