It is believed that any agreement reached will include the option for King to see out the season at Tynecastle.
The fee being discussed is said to be around £200,000, and Hearts need to generate funds as their much-publicised financial struggles continue.
They will be reluctant to part with a product of their academy system, but appreciate that there is little they can do to keep the transfer wolves from their door.
King is an 18-year-old midfielder who is also capable of operating at the back.
He only made his full debut for Hearts in December, but has been attracting attention for some time.
Swansea are believed to have made an approach for his services at the end of the summer transfer window in 2013, but saw an offer knocked back by Hearts' administrators.