The Scotland centre-back is out of contract at the end of the season and has had serious interest north and south of the border.
A host of English Championship clubs were keen on his signature with Blackburn Rovers and Sheffield United at the front of the queue.
Tony Mowbray’s Blackburn side are pushing for promotion to the Premier League which appealed, while Blades boss Paul Heckingbottom is aware of the player’s talents from his time as Hibs boss.
According to the Scottish Sun, Souttar was set to sign a pre-contract agreement with an English side until Rangers’ interest became apparent.
The Ibrox club are in the market for a centre-back with Connor Goldson having still not agreed a new deal with his contract expiring at the end of the season.
They have also been strongly linked with a move for Vitesse Arnhem captain Danilho Doekhi who is also into the final months of his current deal.
Rangers may look to bring Souttar in this month, just as they did with Scott Wright when the midfielder signed a pre-contract agreement when still an Aberdeen player.
Hearts will want to hold onto the 25-year-old as they pursue European football.
Manager Robbie Neilson said it would take a “huge bid” to land him in January.
“From our perspective he’s here until the end of the season, unless we get a big offer we can’t refuse,” he said. “Hopefully within that time we can negotiate a new contract.”
Hearts have signed up Craig Halkett for next season and are close to a deal for Stephen Kingsley.