However, they travel to one of the surprise packages of the competition, the Scottish Championship side having already defeated top-flight opposition to reach the last eight.
Match preview
When being thrashed 5-0 by Rangers, Steven Naismith may have had concerns over how Hearts would react as they bid to remain as the standout best-of-the-rest in the Scottish Premiership.
However, a week on from that embarrassing defeat, Hearts were overcoming Celtic by a 2-0 scoreline at Tynecastle in a game where both sides were on the receiving end of poor refereeing decisions.
Celtic were naturally weakened by an early sending off, yet that should take nothing way from how Hearts were able to get over the line.
They now sit 13 points adrift of their second-placed opponents, yet more importantly, 13 points clear of fourth position as they bid to qualify for the second qualifying round of the Europa League.
Hearts can still achieve continental football through their performances in this competition, away wins over Spartans and Airdrieonians being recorded thus far.
In sharp contrast to Hearts only playing away ties, Greenock Morton have been presented with four successive home ties, overcoming a range of opposition at Cappielow Park.
Bo'ness United and Montrose were defeated by 4-0 and 2-0 scorelines respectively before Morton upset Motherwell last month, running out 2-1 winners.
Dougie Imrie's priorities lie elsewhere with Morton sitting in fourth position in the Championship table, the most realistic target being a spot in the playoff quarter-finals with a 14-point gap to second place.
Nevertheless, Imrie will implore his side to enjoy an occasion such as this one, Morton just two games away from reaching their first Scottish Cup final since the 1947-48 campaign.
Greenock Morton Scottish Cup form:
Greenock Morton form (all competitions):
W
D
W
D
L
L
Hearts Scottish Cup form:
W
W
Hearts form (all competitions):
W
W
W
L
D
W
Team News
Although Imrie may stick with his usual Morton back four, changes could be made in midfield and attack with no goals being scored in their last three outings.
Michael Garrity could get the nod over Jai Quitongo, while Lewis McGrattan is pushing for a recall in midfield.
Barring any fitness issues, Naismith may select the side that defeated Celtic in a game that would have taken place more than a week prior to this one.
Naismith will likely remain with a back three, and Cameron Devlin or Aidan Denholm may be recalled in the engine room if any central-midfield players are handed a breather.
Lawrence Shankland has 27 goals to his name in all competitions this season and should lead the line.
Greenock Morton possible starting lineup:
MacDonald; French, O'Connor, Baird, Strapp; Blues, Power; McGrattan, Crawford, Garrity; Muirhead
Hearts possible starting lineup:
Clark; Sibbick, Kingsley, Rowles; Lembikisa, Nieuwenhof, Baningime, Grant, Cochrane; Forrest, Shankland
We say: Greenock Morton 1-4 Hearts
With all the pressure on Hearts, Morton will feel like they can play with a certain level of freedom to cause one of the upsets of this year's competition. However, we cannot see Hearts faltering on Monday night, the visitors likely to run out convincing victors.