The Ibrox side are hoping to land the Scotland international on a pre-contract agreement but face stiff competition from a number of clubs in England for his signature.
The 25-year-old Tynecastle centre-back, who is out of contract in the summer, is said to have up to 10 clubs chasing him – with a move across the M8 to link up with Giovanni van Bronckhorst among his options.
Blackburn, Sheffield United, Stoke and Middlesborough have also been mentioned, with initial talks involving Souttar's representatives and several clubs, including Rangers, already underway as he looks to secure the best financial and football package on offer.
The Daily Record report that Souttar is expected to reach a final decision by next week, with the former Dundee United stopper ​keeping his options open until then.
Rangers are hoping the lure of guaranteed Champions League football, provided they win the title, can sway the defender’s decision, with current Ibrox centre-back Connor Goldson likely to depart in the summer after stalling on a new contract offer.