The record attendance of 6,000 was set by a Scottish Cup tie against Rangers in 1948.
Hearts meet Falkirk in a league match today which will provide a preview of the forthcoming Scottish Cup tie between the sides.
Morton also played a Scottish Cup tie against Rangers at Love Street in 2000.
Hibs won the first Scottish Cup tie between the clubs, in 1877-78.
Other notable matches between the two clubs include a Scottish Cup tie in 2000.
He made his first senior appearance in a 2009-10 Scottish Cup tie against Hibernian, aged 16.
He made his first team debut in a Scottish Cup tie at Aberdeen on 10 January 2007.
A record attendance of 15,467 was set by a 1954-55 Scottish Cup tie against Celtic.
The biggest ever attendance was 8,123, against Aberdeen in a Scottish Cup tie played on 3 February 1973.
Tynecastle's record attendance was achieved in 1932, when 53,396 attended a Scottish Cup tie against Rangers.