US Open runner-up Medvedev, chasing his fourth ATP Tour title in a breakthrough 2019 season, will meet Germany's Alexander Zverev in Sunday's final.
The three-time Grand Slam winner says he no longer has pain in his hip and it is now a matter of building up fitness so he can play several matches in a row at tournaments.
The 33-year-old Spanish left-hander moved one shy of Roger Federer's all-time men's record 20 Grand Slam triumphs and became the second-oldest New York champion in the Open era after Ken Rosewall in 1970 at age 35.
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.
Grigor Dimitrov, 78th-ranked, delivered a stunning upset of Roger Federer in the US Open quarter-finals, fighting back for a 3-6, 6-4, 3-6, 6-4, 6-2 triumph over the Swiss star who had beaten him in all seven of their prior meetings.
Second seed Rafael Nadal – the winner at Flushing Meadows in 2010, 2013 and 2017 – produced some electrifying moments of brilliance to end the challenge of Croatia's Marin Cilic 6-3, 3-6, 6-1, 6-2.
Champion Novak Djokovic elected to retire in the third set due to a left shoulder injury. He was facing off Stan Wawrinka who was leading 6-4, 7-5, 2-1.
World number two Rafael Nadal advanced to the US Open third round by walkover after Australian Thanasi Kokkinakis withdrew from the tournament with a right shoulder injury.
Osaka will face off against unseeded Pole Magda Linette, who she defeated earlier this year on her way to her second Grand Slam title at the Australian Open.
Williams, who last faced Sharapova at the 2016 Australian Open, improved to 20-2 in career meetings versus the Russian, who has fallen to 87th in the rankings amid an injury-hit 2019 season that included shoulder surgery in February.
Japan's Kei Nishikori is the No 7 seed this year after reaching the semifinals in Flushing Meadows last year.
Serena Williams and Maria Sharapova will open the evening session at Arthur Ashe Stadium with their first-ever NY meeting in a match that may just have the feel of a championship final.
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