The former Manchester United and Sunderland man weighed in with nine goals last term as the club were relegated to the Championship after one season in the top flight.
Following their drop into the second tier, Campbell is one of a host of players linked with moves away, along with the likes of Steven Caulker, Gary Medel and David Marshall.
With newly promoted Leicester City reported to be interested in Campbell, Solskjaer believes that he has done all he can to try to keep the 26-year-old at the club.
"It's not in my court any more, it'll be up to Fraizer to decide what to do," the Norwegian told BBC Wales Sport.
"If he gets offers he might be wanting to stay in the Premier League. But of course I've spoken to him and he knows how much I want him to stay."
Campbell is reported to have a relegation release clause in his contract, with speculation that such caveats could also come into play for Caulker, Medel and Marshall.
While Solskjaer said there was little to report of late in terms of movement, he acknowledged that any such clauses could force Cardiff's hand.
"Not too much has happened that's concrete," he added.
"We do expect probably something to happen with some of them, maybe not all of them. There has been interest shown in our players.
"They've played really well last season, they are talented players and you expect clubs to enquire and ask.
"I can't comment if players have clauses because that's confidential, and not all of them do.
"So I'm working hard to keep the ones I can stop leaving. The ones who've got clauses, I'm pretty tied down really."
