Inter – who have not won a trophy since claiming the Coppa Italia in 2011 – entered the competition in the third round after last year's disappointing Serie A finish.
With the top eight sides last season's Serie A not coming into the competition until the fifth round, Inter's placing of ninth saw them at the risk of a shock loss in the early stages.
But they saw off Cittadella 4-0 in August, as Rodrigo Palacio scored a double in their comfortable win, and they enter their match against Trapani on the back of a good run.
Walter Mazzarri's side have lost just once this season – that being to the second-placed Roma – and sit fourth in the Serie A table.
They were not at their best on Sunday though, drawing 1-1 with Sampdoria at San Siro in a performance that Mazzarri claimed was their poorest of the season.
Inter's injury list is lengthy at the moment, with Diego Milito, Mauro Icardi, Cristian Chivu, Yuto Nagatomo and Walter Samuel all sidelined.
But they still should have far too much about them for Trapani, who lie 13th in Serie B.
Trapani – who were champions of the third tier last season – have taken time to find form in the current campaign but will be buoyed by three successive victories.
The latest of those came on Saturday as Simone Basso's 68th-minute strike gave them a 1-0 triumph over Virtus Lanciano.
Inter are one of eight Serie A sides remaining in the fourth round, with six of them at home.
In the two all top-flight ties, high-flying Verona travel to strugglers Sampdoria on Thursday, while Atalanta take on Sassuolo a day earlier.
Frosinone are the lowest-ranked side still remaining in the competition, sitting fourth in Lega Pro I.
The third-tier outfit are on the road to Avellino on Tuesday, after suffering their first defeat in more than two months on Saturday. Avellino are also in good form and are fourth in the Serie B table.
Other fixtures on Tuesday see Parma host Varese, Bologna take on Siena and Spezia battle Pescara, while on Wednesday, Chievo battle Reggina.
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