Cardiff have said Gabbidon would only go for around £2m, and despite rumours that Everton were set to move in with a bid, it seems unlikely there is any truth in the rumour.
Charlton and West Brom are believed to be among the Premiership sides in the Bluebirds captain.
Relegated Premiership side Southampton have also linked with a mover, but it seems less likely the 25-year old would join another Championship club.
Cardiff still have the task of slashing their wage bill and gaining as much revenue as possible to allow new boss Dave Jones to make new signings before the beginning of the new season.
Admissions have already beenn made at the club that they must sell if they are to bring new blood in.
Winger Jobi McAnuff was the first to leave this summer, joining captain Graeme Kavanagh who towards the end of last season and Rob Earnshaw who left at the beginning of the season.
Club owner Sam Hammam has declared that the club's youngsters are not for sale, however, he could part with Gabbidon for around £2m.