World number one Novak Djokovic will meet 19-year-old Lorenzo Musetti of Italy for a spot in the quarterfinals after he brushed aside 93rd-ranked Ricardas Berankis of Lithuania 6-1, 6-4, 6-1 in third round match on Saturday.
Barty returns to the final after beating No. 5-seeded Elina Svitolina 6-3, 6-3.
Williams will face Canadian Bianca Andreescu for the US Open title as the American sets herself up for another attempt to clinch a record-equaling 24th Grand Slam singles title.
Caroline Wozniacki who was diagnosed with rheumatoid arthritis, an autoimmune disease which attacks the joints, before the US Open says she is proud of how she has been so positive through it all.
US Open winner Sloane Stephens and world number one Simona Halep cruised into the final to set up a rematch of the French Open final.
While Nadal dropped a set to win the mens titles, Elina Svitolina defended the women's title, scorching through to her title in a mere 67 minutes.
Nadal and Djokovic have played each other 50 times, but their last meeting came more than a year ago, when Nadal won in the Madrid Open semifinals. Djokovic leads Nadal 26-24 overall, but Nadal holds a 13-7 advantage on clay.
Serena Williams' hopes of a fifth Olympic gold medal were crushed by Ukraine's Elina Svitolina in a 6-4, 6-3 loss
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