Azarenka was a 6-0, 6-0 winner over Christina McHale in a third-round match and took another step toward defending her tournament title. But even if Azarenka repeats as champion, she will lose the No. 1 ranking should Williams win her quarterfinal match Friday.
Williams, seeded second at Doha, defeated Urszula Radwanska 6-0, 6-3 in 57 minutes Thursday. Williams won 62 percent of the points on serve and 63 off Radwanska's serve. She also finished off 6-of-9 break points.
Her stellar play since losing a first-round match at last year's French Open has put her on the verge of regaining the No. 1 ranking. At 31, she would be the oldest woman to reach the top of the rankings.
Azarenka won 69 percent of the points in her win Thursday. She dropped 10 points in six serving games.
Seven of the top eight seeds advanced to Friday's quarterfinals. Third-seeded Maria Sharapova did so with a 6-3, 6-3 win over Klara Zakopalova. Sharapova had nine double faults and lost her serve three times, however.
Sixth-seeded Sara Errani just got by Daniela Hantuchova 7-5, 5-7, 6-2 in more than 2 1/2 hours while No. 7-seeded Petra Kvitova defeated No. 11 Nadia Petrova 4-6, 6-4, 6-3 and Samantha Stosur, the eighth seed, topped Svetlana Kuznetsova 6-4, 7-5.
In addition to the Radwanska-Wozniacki match Friday, the quarterfinals have Azarenka going against Errani, Sharapova playing Stosur and Kvitova between Williams and the No. 1 ranking.
Caroline Berg Eriksen: Soccer player's wife triggers debate with post-birth selfie
Kate Moss Playboy shoot is classic Playboy, classic Kate