It was current Tynecastle manager Jim Jefferies who signed the player for the Rugby Park side three years ago and he was ear-marked, along with former team-mate Kevin Kyle, as one of the manager's key targets when Jefferies replaced Csaba Laszlo in the capital in January.
Having succeeded in getting striker Kyle, who was out of contract, the battle is now on to convince Kilmarnock chairman Michael Johnston to accept the figure of around 350,000 being offered for the 23-year-old Bryson who recently signed a contract-extension with the Ayrshire club and still has two years of the deal to run.
But Kyle hopes Hearts get their man and has assured the club that any fee will be money well spent. "If Hearts got Craig Bryson in I would welcome him and I am sure the fans would as well," said Kyle following his arrival in Gorgie.
"He is a cracking little player. Kilmarnock will be desperate to hold on to him because he is probably their most prized asset right now and it may be difficult to lure him away, but if Jim could get him it would be a magnificent coup and it would benefit Hearts enormously. The transfer market is a tricky thing but Jim has plenty experience of it and I am sure the owners of the club will try and help him the best they can."
Owner Vladimir Romanov has promised to give Jefferies financial backing to bring in new players, provided they are young enough and have the potential to secure a high sell-on value. The club believes Bryson is a player who falls into that category and Kyle agrees. He would love to see the squad further strengthened following his own signing and the arrival of defender Darren Barr.
"He has been talked about in Scottish football for the last season or two and rightly so because he is a fantastic little midfielder. He has great energy and he is a box to box player, more of an attacker than a defender but he can certainly defend as well. He has got class and skill and ability and he has a great range of passing," said Kyle.