The Jam Tarts announced a two-and-a-half year extension for centre-back partner Craig Halkett on Tuesday.
Kingsley, The Scotsman understands, is on the verge of following suit.
The 27-year-old has been one of the club’s most consistent performers since arriving on a free contract after leaving Hull City. He has scored nine goals in 48 appearances.
Back in September, Kingsley answered “absolutely” when asked if he was keen to extend his stay at Tynecastle Park beyond his current contract which runs until the summer.
Manager Robbie Neilson confirmed in November that the club had made an offer for the player.
He said: "Hopefully we can get Stephen tied up because he has been one of our better players this season, and probably last season as well in terms of consistency.”
A new deal was hinted at by sporting director Joe Savage following the announcement of Halkett’s contract extension.
He said: “It’s great news ahead of our return to action, and I’m hopeful that we’ll have even more good news in the coming days.”
There are a handful of Hearts players on expiring deals, including Michael Smith and Peter Haring.
One player who has all but played his last game for the club is Jamie Walker. The attacker signed a loan deal with Bradford City and posted a farewell message on social media.
He said: “I wish the club, the supporters, my team-mates (a great bunch of lads) and the staff all the best for the rest of the season and hopefully I’ll be bringing my son to watch European football next season.”