Serena Williams issued an ominous warning to budding young tennis players on Monday - her three-year-old daughter loves the game and will soon take it up.

"She's a fan of tennis, she loves watching, she comes out and watches me practice," Williams said of Olympia after moving into the US Open last eight.

The toddler was seen smiling and waving at her mother during Williams' third-round win against Sloane Stephens inside Arthur Ashe Stadium on Saturday.

"She loves when I practice on this court because she can sit and watch," Williams said after beating Greece's Maria Sakkari in the last 16 on Arthur Ashe.

"She's three now. I'm like, 'I gotta put my career aside and get her a racquet in her hand.

"Seriously, that's what I told my husband yesterday. I'm like, 'We got to get a racquet in her hand," Williams added.

The 23-time Grand Slam winner defeated Sakkari 6-3, 6-7 (6/8), 6-3 to set up a last-eight clash with Tsvetana Pironkova, who is also a mother.

The Bulgarian is playing in her first tournament in three years after giving birth to a son.

Results from US Open tennis championship in New York on Monday (x-denotes seed)

Men's singles

Fourth round

Andrey Rublev (RUS x10) bt Matteo Berrettini (ITA x6) 4-6, 6-3, 6-3, 6-3

Alex De Minaur (AUS x21) bt Vasek Pospisil (CAN) 7-6 (8/6), 6-3, 6-2

Dominic Thiem (AUT x2) bt Felix Auger-Aliassime (CAN x15) 7-6 (7/4), 6-1, 6-1

Daniil Medvedev (RUS x3) bt Frances Tiafoe (USA) 6-4, 6-1, 6-0

Women's singles

Fourth round

Tsvetana Pironkova (BUL) bt Alize Cornet (FRA) 6-4, 6-7 (5/7), 6-3

Serena Williams (USA x3) bt Maria Sakkari (GRE x15) 6-3, 6-7 (6/8), 6-3

Victoria Azarenka (BLR) bt Karolína Muchová (CZE x20) 5-7, 6-1, 6-4

Elise Mertens (BEL x16) bt Sofia Kenin (USA x2) 6-3, 6-3