Goal understands the Gunners have opened talks with the relegated side regarding the 29-year-old and are willing to pay up to £8 million for his services as they look to replace new Swansea keeper Lukasz Fabianski.
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"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."
Despite Arsene Wenger taking the first steps to bring Marshall to the Emirates Stadium Solskjaer insists Cardiff have yet to receive a substantial offer for the player.
He added: "There have been no big bids for David yet and to be honest I don't want any big bids for him either."