Romania's Halep, a French Open runner-up for the second time in June, can overtake top-ranked Czech defending champion Karolina Pliskova.
Romania's Halep, a French Open runner-up for the second time in June, can overtake top-ranked Czech defending champion Karolina Pliskova by winning the title in the final major tuneup for the US Open.
"To be number one in the world I think is a big thing," Halep said. "If I deserve the place, for sure I will win it."
Pliskova, who faces Italian qualifier Camila Giorgi in a later third-round match, must also fight off Ukraine's Elina Svitolina, who won her fifth WTA title of the year last week in Toronto to reach a career-high fourth in the rankings.
Svitolina, who faces Germany's Julia Goerges later, beat Halep in the Toronto final for her fifth title of the year.
"I had a tough run," Halep said. "I'm not playing my best tennis in this moment but I'm still winning. I win against the 15th player in the world playing not like I want and I'm really happy about it."
Halep, who won her 15th career title earlier this year at Madrid, next faces either British seventh seed Johanna Konta or Slovakian 11th seed Dominika Cibulkova.
Serena Williams, expected to give birth next month, had been atop the rankings after winning her 23rd career Grand Slam title at the Australian Open while pregnant.
Also reaching the quarter-finals was Wimbledon champion Garbine Muguruza. The sixth-ranked Spaniard saved three match points in downing 17th-ranked American Madison Keys 6-4, 3-6, 7-6 (7/3).
After a two-hour rain delay at 2-2 in the third set, they exchanged breaks before Keys broke again in the 11th game to serve for the match.
But Keys, who improved to 3-0 against Muguruza with a Stanford semi-final win two weeks ago, squandered three match points with errant shots and sent a backhand long on break point to force a final tie-breaker which Muguruza led from start to finish.
"You never knew who was going to win until the last point," Muguruza said. "I had match points against me so it was really hard. I battled back and I'm glad things went my way."
Muguruza has a last-eight date with eighth seed Svetlana Kuznetsova. The 32-year-old Russian, a two-time Grand Slam champion, eliminated Spain's Carla Suarez Navarro 6-2, 6-4.
Spaniard Rafael Nadal, the 15-time Grand Slam champion assured of returning to world number one next week, faces countryman Albert Ramos-Vinolas in a night match. Nadal could meet Australian Nick Kyrgios in Friday's quarter-finals.
Nadal was assured of advancing past injured Andy Murray for the top spot when Roger Federer withdrew from Cincinnati with a back injury.
Bulgaria's 11th-ranked Grigor Dimitrov won his first ATP match in six tries against Argentina's Juan Martin del Potro by 6-3, 7-5. Dimitrov's most recent setback had been in the first round this year at Rome.
"I lost five times against him. I know what to expect now," Dimitrov said. "I really needed that match. There are still a few guys I haven't conquered yet but that will come in time."
Dimitrov next faces Japan's Yuichi Sugita, who rallied past Russian Karen Khachanov 6-7 (0/7), 6-3, 6-3.