Copil, the World No 93, stunned second seed Alexander Zverev of Germany 6-3, 6-7 (6), 6-4 in the day's other semi-final.
The Serb was knocked off top spot by Rafael Nadal on Monday despite winning the Paris Masters last week and will need a strong finish alongside the River Thames to snatch it back.
Medvedev squandered a two-set lead against Nadal in their previous meeting in the Melbourne final in January, allowing the Spaniard to clinch his 21st Grand Slam title.
World number one Novak Djokovic produced a majestic display in a 7-5, 6-3 victory over American John Isner on Thursday to extend his winning streak in Asia.
World number one Novak Djokovic continued his dominant form in Tokyo to storm past Frenchman Lucas Pouille 6-1, 6-2 and into the semi-finals of the Japan Open.
Images from Day 7 of the 2020 US Open in New York on Sunday.
Images from Day 6 of Wimbledon at the All England Lawn Tennis and Croquet Club in London, on Saturday.
Images from the Australian Open in Melbourne on Thursday.
American Sloane Stephens kept intact her streak of never having lost a final as she defeated Latvia's Jelena Ostapenko 7-6(5), 6-1 to claim the Miami Open title on Saturday.
Images from Day 6 of the US Open in New York, on Saturday.
Former world number one Andy Murray's comeback from a year-long injury layoff was ended by Nick Kyrgios at Queen's Club on Tuesday but only after the Briton produced a typically dogged display of resistance.
Take a look at six of the up-and-comers who could upset the top names at the year's first Grand Slam.
'Irresponsible' Djokovic panned for Adria Tour misadventure.
Rafael Nadal suffered a shock 7-5, 6-3 defeat by Austrian Dominic Thiem in the Madrid Open quarter-finals on Friday, a result that means the Spaniard will lose the world number one ranking.
IMAGES from Day 5 of the French Open at Roland Garros in Paris on Thursday.
Novak Djokovic eased into the last four with a commanding performance over Japan's Kei Nishikori but world number two Daniil Medvedev bowed out of the men's singles at the Tokyo Olympics.
World number one Simona Halep saved three match-points before edging past an inspired Lauren Davis 4-6, 6-4, 15-13 in a thrilling marathon to reach the fourth round of the Australian Open on Saturday.
Thiem, Wawrinka stay positive after defeats on return
Ash Barty beat eighth seed Kiki Bertens of the Netherlands 6-3, 3-6, 7-6(7) in Beijing to seal her spot in the final of the China Open on Saturday.
Raonic advances, Sock ousted in Rogers Cup
Nadal will not be playing at Wimbledon, having decided to rest, while eight-time champion Federer is showing signs that, at 39, the window of opportunity to win one more Slam is closing.
World number one Novak Djokovic extended his winning run in Asia when he strolled into the last-16 of the Shanghai Masters.
Frenchman Benoit Paire confirmed that he had tested positive for COVID-19. This has meant they have been barred from using any on site player facilities.
Images from Day 4 of the Wimbledon tennis championships played in London, on Wednesday.
It took the COVID-19 pandemic to put a brake on Djokovic's 18-match unbeaten run in 2020, as he won the ATP Cup with Serbia, an eighth Australian Open title and a fifth Dubai Tennis Championships before the season came to a halt in early March.
Roger Federer began his latest quest for a 100th ATP title with a battling 6-4, 3-6, 6-1 win over Philipp Kohlschreiber in the first round of the Dubai Duty Free Tennis Championships on Monday.
Dominic Thiem and Stefanos Tsitsipas won their respective singles matches before Roger Federer and Alexander Zverev teamed up in the doubles to beat Jack Sock and Denis Shapovalov
With two men's five-set matches having caused a log-jam, Muguruza and Konta did not begin their match until 12:30 am local time (1330 GMT) on Friday at Margaret Court Arena, a record late start at the Grand Slam.
Defending champion Rafael Nadal thrashed Austrian fifth seed Dominic Thiem 6-0, 6-2 to reach the semi-finals of the Monte Carlo Masters for a record 14th time on Friday.
Victory meant Nadal set the record for the longest Davis Cup winning streak in combined singles and doubles rubbers with 23 matches in a row.
Novak Djokovic is chasing a record-extending eighth Australian Open title, while Dominic Thiem will bid to become the first man born in the 1990s to win a major title.
Murray to practise with Djokovic in Rome before return from injury.
Wimbledon champion Garbine Muguruza notched up her first ever victory on Arthur Ashe Stadium with a 6-0, 6-3 win over American Varvara Lepchenko at the US Open on Monday.
Djokovic had his personal best winning sequence of 30 consecutive sets ended as Cilic played almost flawlessly aggressive tennis to draw first blood in a magnificent duel.
Twice Shanghai Masters champion Roger Federer was far from his best against Spain's Roberto Bautista Agut but did enough to reach the quarter-finals with a 6-3, 2-6, 6-4 win on Thursday.
Top seed and defending champion Roger Federer battled past in-form Russian Daniil Medvedev 6-4, 4-6, 6-4 to reach the last-16 at the Shanghai Masters on Wednesday.
Big-serving Canadian Milos Raonic fired 19 aces to topple Sam Querrey 6-4, 6-4 and set up a second-round clash with either Dan Evans or Russian 10th seed Andrey Rublev.
Grigor Dimitrov stood one win away from the biggest title of his career on Saturday after ending his Jack Sock jinx to reach the final of the season-ending ATP Finals at the 02 Arena.
Bulgarian's Grigor Dimitrov claimed a first career win over former world number one Rafael Nadal at the eighth time of asking to reach the semi-finals of the China Open on Friday.
Greek third seed Stefanos Tsitsipas battled back from a set down to beat Serbia's Filip Krajinovic 3-6, 6-4, 6-4 in the quarter-finals of the Swiss Indoor championship on Friday. The win took Tsitsipas into his 10th semi-final of the season where he will face top seed Roger Federer, who was handed a walkover after Swiss compatriot Stan Wawrinka withdrew before their quarter-final with a back issue.
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