Although Hughes is willing to let the 37-year-old go, he insists Fulham must first find a replacement and Given has emerged as his prime target.
The Manchester City keeper has seen Joe Hart reclaim first-choice status after returning from last season's loan spell at Birmingham.
City boss Roberto Mancini would like Given to stay and provide competition for Hart but at 34, the Ireland international wants regular first-team football.
That could come at Craven Cottage, where former City boss Hughes succeeded Roy Hodgson this summer.
"Mark Schwarzer has made it clear that he would like to go to Arsenal so we are trying to look at other options," said the Fulham boss. "I'm always interested in quality players and Shay Given is a quality player. Make of that what you will.
"But I have said that if we can't get anyone else in, Schwarzer will have to stay."
Hughes signed Given from Newcastle for a cut-price £8m in January 2009 and he soon proved his class in the City goal.
However, Given lost out to Hart in the battle to start the season as City's first-choice keeper and the new England number one looks set to hold onto his place having been in sparkling form.