Four-time Djokovic was leading 6-1, 1-2 when play was halted for the night.
"We officially announce that due to the heavy rain, the match between #Djokovic and #DelPotro will resume tomorrow. Time TBD," said a statement on the tournament's Twitter account.
The winner will face Dominic Thiem, who knocked out Rafael Nadal, in the semi-finals which will also be played on Saturday.
Results on the sixth day of the Rome Masters on Thursday (x denotes seeding):
Dominic Thiem (AUT x8) bt Rafael Nadal (ESP x4) 6-4, 6-3
Alexander Zverev (GER x16) bt Milos Raonic (CAN x5) 7-6 (7/4), 6-1
John Isner (USA) bt Marin Cilic (CRO x6) 7-6 (7/3), 2-6, 7-6 (7/2)
Garbine Muguruza (ESP x3) bt Venus Williams (USA x9) 6-2, 3-6, 6-2
Elina Svitolina (UKR x8) bt Karolina Pliskova (CZE x2) 6-2, 7-6 (9/7)
Kiki Bertens (NED x15) bt Daria Gavrilova (AUS) 6-3, 6-3
Simona Halep (ROM x 6) bt Anett Kontaveit (EST) 6-2, 6-4