Kazakhstan's Rybakina, who added the French Open title to her 2022 Wimbledon crown this year, is into the US Open last eight for the first time in her career.
The meeting with 2024 Olympic gold medallist Zheng will carry extra weight. If Rybakina reaches the semi-finals, she is assured of supplanting Aryna Sabalenka atop the world rankings, even if the Belarusian wins her third straight US Open title.
"Of course I want it," Rybakina said. "I don't know anyone who wouldn't wish this."
But she added: "I try to approach the same, fight to the end (in) my next match - let's see what will happen."
Rybakina needed just 66 minutes to get past Osaka, firing 29 winners and saving the only break point she faced.
Osaka fought off a pair of match points in the penultimate game, but ultimately the unforced errors that have plagued her all tournament proved too much to overcome.