Supporters continue to buy up seats in their numbers after a general sale commenced for the tie against Inverness Caledonian Thistle at Hampden Park on Saturday, April 13.
Tickets went on sale - initially to season ticket holders - at 9am last Friday morning and have been moving steadily since. They reached the 13,000 mark earlier today.
Hearts have been allocated 25,000 seats at the National Stadium and are favourites to win in their second cup semi-final of the season.
Inverness, managed by the Tynecastle Park legend John Robertson, play in Scotland's Championship and are regarded as underdogs against Premiership opponents.
Hearts fans interested in attending the semi-final can buy tickets via www.heartsfc.co.uk, or by calling 0333 043 1874 (option 1 then 1), or in person at the Hearts Ticket Centre.
Tickets are priced £30/£20 for adults, £15/£10 for concessions.