In another interesting quarter-final match-up, No. 1 seed from Australia, Ashleigh Barty could face Aryna Sabalenka in a rematch of the Stuttgart and Madrid Open finals.
Earlier in the tournament, on Wednesday, the unseeded Bopanna and Shapovalov had taken down top seeds Juan Sebastian Cabal and Robert Farah in straight sets.
Zverev defeated Nadal 6-4, 6-4 in the quarterfinals, picking up his third consecutive win over the 20-time grand slam champion — but first on clay.
Thiem was able to break the tall American on his first chance in the second set and then once more in the third, defeating Isner 3-6, 6-3, 6-4.
Nadal struggled at times against the aggressive style of the 76th-ranked Popyrin but was able to stay in control for most of the match on the Magic Box center court.
Bopanna and Shapovalov will take on German duo of Alexander Zverev and Tim Puetz in the quarter-finals.
Nadal, seeking his sixth title in Madrid and looking to regain top form ahead of the French Open, was never threatened by Alcaraz, who turned 18 on Wednesday.
The fourth-ranked Austrian was coming off early round losses in Doha and Dubai and felt like he needed a break.
The ITF had been working with the Hungarian government and the HTA to explore "all feasible options" to reschedule this year's finals.
The rising Czech captured her third win over a top-5 player this season, and her 10th top-5 win in her career overall, with her one-hour and 49-minute victory over reigning Australian Open champion Osaka.
The Spaniard was scheduled to face Sloane Stephens in the first round.
The top-seeded Zverev, who won the tournament in 2017 and 2018, hit six aces, saved three of the five break points he faced and converted five of his own against his 89th-ranked opponent.
Instead, under a plan organizers “intend to make a permanent part of the schedule,” the fourth round will be split between Sunday and Monday next year, All England Club chairman Ian Hewitt said during a video conference.
Nadal roared out to a 5-1 lead before Carreno pulled back a break. Carreno then had three break chances on Nadal’s next service game, but the latter saved them all and closed out the set.
Nadal broke Nishikori twice in the decisive set, converting his second match point by driving a forehand winner down the line.
Djokovic easily overcame the South Korean in less than 90 minutes to qualify for the quarterfinals of the ATP 250 tournament.
Nadal had to overcome a slow start to defeat Ilya Ivashka of Belarus 3-6, 6-2, 6-4 at the clay-court tournament.
Nagal lost to Frenchman Pierre-Hugues Herbert, a top-100 player, 5-7, 0-6 after failing to convert two break chances in the first set when the match was evenly poised at 5-all and Nagal was leading 15-40.
Fourth-seeded Tsitsipas was in dominant form as he edged 4-3 ahead of the sixth-seeded Russian in their career meetings, having also beaten him on clay last year in the French Open quarterfinals.
The 136th-ranked Nagal, who is aiming to get into the top-100 on the ATP rankings, will now take on Italy's Thomas Fabbiano for a place in the main draw of the tournament, which is one of the most important events in Spanish tennis.
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