Osaka, who is now coached by Wim Fissette, who was earlier coaching Zheng but left the Chinese player last year to work with the Japanese player for a second time in their careers.
Zheng, ranked No 7, will play the winner of the match between American Coco Gauff, the No 3 seed, and Spaniard Paula Badosa next.
The 21-year-old Chinese won against American Shelby Rogers (6-2, 6-0) and No 29 seed Linda Noskova (3-6, 6-1, 6-2) earlier in the week.
In the previous rounds of the tournament, Osaka, ranked No 173, beat Frenchwoman Clara Burel (7-6 (2), 6-1), Ukrainian Marta Kostyuk, the No 19 seed (6-3, 6-2) and Russian Daria Kasatkina, the No 10 seed (6-3, 6-3).
Roma WTA 1000, other round of 16 results (Foro Italico, clay, USD 5,509,771):
- Elina Svitolina vs. Aryna Sabalenka
- Maria Sakkari vs. Victoria Azarenka
- Danielle Collins vs. Irina-Camelia Begu
- Iga Swiatek vs. Angelique Kerber
- Rebecca Sramkova vs. Jelena Ostapenko
- Coco Gauff vs. Paula Badosa
- Madison Keys vs. Sorana Cirstea
