King is regarded as one of the best young players to emerge in Scotland in recent years and is tipped for a big future.
The 17-year-old has represented Scotland at Under 18 and Under 19 level and is held in high regard within the corridors of power at Tynecastle.
A number of English Premier League clubs are thought to have been alerted to King's talent and have been monitoring his progress.
Swansea are believed to have firmed up their interest in King with a bid for the midfielder, but Hearts turned down the offer.
Despite being in administration, Hearts were able to fend off the bid in King like they did with Nottingham Forest's offers for Jason Holt.