The Tynecastle club have already had one bid for the 21-year-old knocked back by his club, Botev Plovdiv, after offering £100,000 plus a 20 per cent sell-on fee.
Gamakov, who was runner-up for the Young Player of the Year award in his homeland last season, is also thought to be attracting interest from Italian Serie A outfit Palermo. He has one year left to run on his contract and Plovdiv, who are in a precarious financial position, are keen to cash in on one of their prize assets during this transfer window.
Nevertheless, Hearts have been told that they will need to substantially up their offer, with a fee of €500,000 (£350,000) quoted by sources in Bulgaria.
“The [first] offer has not been accepted, but Hearts have not given up,” confirmed Gamakov’s representative, Milen Georgiev. “There is also interest from other teams.”
The playmaker, who cuts an imposing figure at 6ft 2in, is considered one of the most promising talents in Bulgarian football and has represented his country at every youth level up to under-21. He received his maiden call-up for the senior side last November for a Euro 2016 qualifier against Malta, but was an unused substitute.