Clijsters needed slightly longer against Belarus' Tatiana Poutchek, but was just as emphatic in a 6-2, 6-1 55-minute workout.
Williams' win sets her up for a fourth round meeting with Greece's Eleni Daniilidou, who defeated French 16th seed Nathalie Dechy 6-4, 6-3.
The American superstar, who is chasing a "Serena Slam" here after her three Grand Slam victories, said afterwards she believed big sister Venus was anxious to dethrone her.
"She's never sat down and told me, 'Serena I'm going to take No.1 away from you," Williams said. "But I'm sure that's what she wants to do.
"She's working hard, she's playing really well. She is just on a different level from a lot of the girls and everyone else -- including me sometimes."
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