Earlier on Monday, Rybakina overpowered 13th seed Osaka to win 6-1 6-4 on Louis Armstrong Stadium and set up a quarter-final against Zheng.
Kazakhstan’s Rybakina will become the world’s leading player for the first time if she reaches the semi-finals in New York, even if top-ranked Aryna Sabalenka goes on to defend her title.
China’s Zheng came from 5-0 down in the first set of her fourth-round match to beat former champion Iga Swiatek.
“Of course I want it, I don’t know anyone who won’t want this,” Rybakina said of the number-one ranking.
“I tried to approach it the same way and do everything I can, and fight to the end of my next match.
“If I serve as well as I did today, I think I can do well. Let’s hope it goes my way.”
Rybakina, 27, dropped only four points in a near flawless performance on serve against four-time Grand Slam champion Osaka, who said after the match she was struggling with pain in her leg.
She faced the 28-year-old full of confidence, having not dropped a set in her opening three matches at the tournament, and took control of the contest in a dominant first set.
Reigning Australian Open champion Rybakina opened up a double break over Osaka, before sealing the set in just 21 minutes.
Osaka showed greater resistance early in the second set but, after failing to convert her only break point of the match, Rybakina soon seized control again.
She broke to go 4-3 up, before safely seeing out the rest of the match, with Osaka unable to contend with her opponent’s deep groundstrokes.
Rybakina sealed the match inside 66 minutes with her seventh ace of a match where she hit a total of 29 winners to Osaka’s nine.
Asked about the gap of almost six years since her last major win, at the 2021 Australian Open, Osaka said: “I thought about it, and if I win another Grand Slam, it’s not going to change my life.
“That’s kind of crazy that I can say that, because I remember the first Slam that I won, it was, like, a huge whirlwind.”