Either Halep or Petra Kvitova can replace Naomi Osaka as world number one at the end of the tournament.
Czech Kvitova beat France’s Caroline Garcia 6-3 6-3 to reach her fifth WTA quarter final of year.
The next seed will play Ashleigh Barty following the Australian fought back to beat Kiki Bertens 46 6-3 6-2.
Osaka will remain on top of the rankings unless Halep or Kvitova hit the final.
If both players make the ultimate, then a winner will claim the number one standing.
Romanian Halep was world number only for 4-8 weeks before being substituted by Osaka in January, while twotime Wimbledon champion Kvitova never held the top ranking.
Canadian teen ager Bianca Andreescu’s 10-match unbeaten run came to an end because she retired with trauma.
The 19-year-old trailed Estonia’s Anett Kontaveit 6-1 2-0 when she withdrew with the right shoulder problem.
Hsieh Su-wei, who knocked out top seed Osaka on Sunday, continued her good form using a three-set win over Caroline Wozniacki.
The Taiwanese beat the former world number one 6-3 6-7 (0-7) 6-2 and can play Kontaveit from the previous eight years.