The 25-year-old travelled south to discuss pre-contract talks with Blackburn Rovers and Sheffield United last week, but the Gers have now put an offer on the table.
The Hearts defender is seriously giving consideration, and he will make the final decision in the coming days.
The Gers are confident of signing the defender whose contract at Hearts is set to expire at the end of the season.
Souttar has a big decision to make here.
If he joins either Blackburn or Sheffield United, he has the possibility of reaching the Premier League if either of the clubs achieve promotion.
On the other hand, he can win titles with Rangers, while the club can offer regular European football at the Champions League level.
Obviously, clubs in England can offer him better pay packages but the lure of European football can be too hard to put down.
In fact, Rangers appear to have offered him a lucrative deal as well.
The Gers have the money burning a hole in their pockets following the departure of Nathan Patterson to Everton, with Rangers pocketing a club-record fee from his transfer.
Hearts offered one of their largest contracts to stay at
Tynecastle Park, but they are resigned to losing the Scotland international.
The Tynecastle club are also in a dilemma – whether to sell him now and pocket some money or keep him till the end of the season and watch him leave for nothing.