The club have pinpointed Robertson to lead the club back into the Ladbrokes Premiership following their relegation under Richie Foran.
Robertson, who works in a club ambassadorial role at Tynecastle, previously managed Inverness between 2002 and 2004.
He is believed to be in talks with Hearts owner Ann Budge to be given permission to take on the role after impressing the Caley Jags in an interview, according to the Press and Journal.
In recent week speculation suggested that former Inverness player and current Aberdeen youth coach Paul Sheerin would be appointed with Maurice Malpas and Danny Lennon the other candidates making up the shortlist.
However, the club have turned to Robertson who steered the club to the top-flight for the first time in their history, as well as a famous victory in the cup against Martin O’Neil’s Celtic.
He then replaced current Hearts director of football Craig Levein as hearts manager in 2004, but his reign was ended by Vladimir Romanov in 2005.
Since then he has had spells as manager of Ross County, Livingson, Derry City and East Fife.