Williams has never dropped a set in four previous meetings with Kanepi, the last of which was at the 2014 U.S. Open, but the Estonian is high on confidence after eliminating top seed Simona Halep in the first round.
Kanepi has benefited from taking Halep's place in the draw and also won her second and third round matches in straight sets, but Williams showed during her demolition of her big sister Venus in the last round that she is close to her best.
Nadal is up against Nikoloz Basilashvili in the first match on Arthur Ashe and dropped only one game in three sets the only other time he met the Georgian -- at the 2017 French Open.
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Stephens, fresh from an impressive victory over the hard-hitting Victoria Azarenka, takes on 15th seed Elise Mertens of Belgium, who beat the American third seed in straight sets at the Western and Southern Open in Cincinnati last month.
Big-serving Argentine Juan Martin del Potro, who has been in cruise control so far, faces 20th seeded Croat Borna Coric, while over at Louis Armstrong Stadium, clay court specialist Dominic Thiem, the ninth seed, faces big serving Kevin Anderson.
American John Isner is up against Canada's Milos Raonic in another clash of big servers, while eighth seeded Czech Karolina Pliskova takes on Australia's Ashleigh Barty.
(Reporting by Simon Jennings; Editing by Toby Davis)
