Only Juventus have won Italy's domestic cup more times than Roma's tally of nine titles, while Genoa's solitary triumph in this competition came all the way back in 1937.
Match preview
Indeed, having lifted the inaugural Europa Conference League last season, Jose Mourinho will be keen to add to both his personal and the club's collective impressive haul of trophies come the end of the current campaign.
With runaway league leaders Napoli 13 points clear of the Yellow and Reds, any prospects of lifting Serie A for the first time in 22 years have surely already evaporated, with top four and Champions League qualification the best that Roma can hope for when it comes to achieving something tangible on that front.
Two late strikes from Roger Ibanez and Tammy Abraham away to the reigning champions AC Milan on Sunday ensured that their opponents and Juventus remained within six points of them in third and second spot respectively, though, and such an impressive result - especially given the circumstances - should boost confidence ahead of hosting Genoa at Stadio Olimpico this week.
Mourinho's side remain in the hunt for the Europa League, too, which not only provides another opportunity of collecting European silverware, but directly qualifying for the Champions League as well. Austrian champions RB Salzburg await them in a double header next month.
Despite suffering from some patchy form at times this season, then, there is plenty for Roma to play for between now and the end of the season, with a potential meeting against Napoli in the Coppa Italia quarter-finals awaiting them should they manage to swat Genoa aside as expected on Thursday.
Having been relegated from Serie A last season, Genoa have already had to come through two rounds in order to progress to this stage of the Coppa Italia.
Fellow Serie B sides Benevento and SPAL were defeated 3-2 and 1-0 respectively, with Albert Gudmundsson getting himself on the scoresheet three times across those two victories.
As such, the Griffin could reach the quarter-finals of Italy's domestic cup for the first time since the 1991-92 season, when they were defeated 4-1 by the eventual champions Parma.
However, winning Serie B for a record-extending seventh time undoubtedly remains Alberto Gilardino's side's priority for this campaign, as they will not want to be stranded below Italy's top tier for an extended period of time as proved to be the case after their previous relegation in 1995. They currently occupy third place in Italy's second tier, with Frosinone and Reggina six and three points ahead of them respectively in the automatic promotion positions.
Defeating one of Italy's giants would be a major scalp for Gilardino at this point in his managerial career, though, with the former AC Milan striker still aiming to make a name for himself in this sphere after previous spells with Rezzato, Pro Vercelli and Siena.
Roma Coppa Italia form:
Roma form (all competitions):
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L
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D
Genoa Coppa Italia form:
W
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Genoa form (all competitions):
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W
D
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W
Team News
Roma are expected to remain without Georginio Wijnaldum, who fractured his tibia shortly after arriving on loan from Paris Saint-Germain last summer. The former Liverpool midfielder could return to action later this month.
Rick Karsdorp was left out of the travelling squad entirely for the weekend's trip to Milan, with the right-back unlikely to play for the club again after falling out with his manager Mourinho.
The former Chelsea boss may opt to make several changes for Thursday's cup tie, with the likes of Nemanja Matic, Andrea Belotti and Stephan El Shaarawy potentially named in the starting XI after coming off the bench against Milan last time out.
Genoa, meanwhile, may make the trip to the capital without Manolo Portanova (personal reasons), Marko Pajac (knee surgery) and Caleb Ekuban (Achilles tendon rupture).
Gilardino's side have not been involved in competitive action since Boxing Day, so the 40-year-old could opt to name a strong side despite facing Venezia in Serie B on Monday.
Roma possible starting lineup:
Svilar; Kumbulla, Smalling, Ibanez; Celik, Camara, Matic, Spinazzola; El Shaarawy, Shomurodov; Belotti
Genoa possible starting lineup:
Martinez; Heft, Bani, Dragusin, Sabelli; Frendrup, Strootman, Jagiello; Aramu, Puscas, Gudmundsson
We say: Roma 2-0 Genoa
Although Mourinho may choose to make several selection changes given his side's push for the top four and involvement in European football this season, Roma should still have too much strength in depth for their second tier opponents and be able to secure safe passage through to the quarter-finals as a result.