Around 1300 tickets have been bought by fans already and it is thought that number could swell as kick-off draws near.
That is despite initial protests from Inverness supporters against the choice of venue and early kick-off time. The cup tie is scheduled to start at 12.15pm and fans travelling from the Highlands were concerned that it put them at an undue disadvantage, not least since Hearts fans needed only to cross the city.
Indeed, one fans group appealed to the Scottish Professional Football Association about the issues and were told that the league body would endeavour to ensure additional public transport was made available on the day of the game.
There is one figure with a fondness for Inverness who will not be in attendance, though. Terry Butcher left his position as manager of the club earlier this season to take charge at Hibs but the Englishman has resolved to stay away from the cup semi-final.
"I don't want to be there," said Butcher, who was in charge of Inverness when the club faced Hearts at the same stage of the competition last season. "That's a personal choice for me."