The central defender is out of contract at the end of the season and is expected to decide next week to accept an offer from Rangers, or move to the English Championship, where he is being chased by Blackburn Rovers, Stoke City, Nottingham Forest, Sheffield United and Cardiff City.
The Tynecastle outfit have all but given up hope of persuading Souttar to stay in Edinburgh despite making him an attractive deal, with a pre-contract agreement with one of the aforementioned clubs looking increasingly likely.
However, if the Scotland centre-back takes Rangers’ offer, the Sunday Mail are reporting that Souttar would want to make the move along the M8 now rather than wait until the summer.
Hearts are pushing for European football this season and currently sit third in the cinch Premiership, with the 25-year-old an integral part of the their defence.
Having just recouped a sizeable fee that could reach £16million for Nathan Patterson after he was sold to Everton earlier this week, Hearts know that Rangers have money to spend and could demand a reasonable fee for an early release.
Rangers manager Giovanni van Bronckhorst has earmarked Souttar as a replacement for Connor Goldson, who is set to leave Rangers at the end of the season when his contract expires. Fellow centre-halves Jack Simpson and Nikola Katic have also been linked with moves away, while Rangers are also interested in Dutch defender Danilho Doekhi.