Uruguayan midfielder Gargano and Japanese defender Nagatomo were both withdrawn in the second half of Sunday's 2-0 Serie A defeat at Cagliari, with medical examinations confirming leg injuries for the duo.
Inter must overturn Roma's 2-1 first-leg advantage in Milan to reach May's final against Lazio, but head coach Andrea Stramaccioni will be without Joel Obi, Gaby Mudingayi, Diego Milito, Luca Castellazzi, Cristian Chivu, Dejan Stankovic, Rodrigo Palacio, Antonio Cassano, Ibrahim M'Baye, Gargano and Nagatomo.
A statement published on the club's website read: "Medical examinations carried out after yesterday's match against Cagliari have confirmed damage to Yuto Nagatomo's left meniscus (knee joint).
"For Walter Gargano, who left the field after eight minutes of the second half, there is second-to-third degree damage to the rectus femoris muscle of his left thigh."
No prognosis for recovery was issued but the injuries should rule the pair out of Wednesday's game and forthcoming league ties against Palermo, Napoli and Lazio.
The absence of Palacio, Milito and Cassano has been particularly damaging to Inter's league form, with 35-year-old Tommaso Rocchi currently operating as the club's only fit senior striker alongside attacking midfielder Ricky Alvarez.
Sunday's defeat to Cagliari at the Stadio Nereo Rocco in Trieste leaves Inter seventh in the Serie A table with third-placed AC Milan and the final Champions League spot nine points away with six games remaining.