Rafael Nadal swept into the Madrid Open quarter-finals on Thursday with a 6-3, 6-3 victory over Alexei Popyrin, while Ashleigh Barty booked her spot in the women's final by ending the run of wildcard Paula Badosa.

Nadal's path to a possible sixth Madrid title was cleared of second seeded Daniil Medvedev and fourth-seeded Monte Carlo champion Stefanos Tsitsipas who were both shock third round losers.

Nadal saved five of six break points in a challenging opening set as he faced the 76th-ranked Popyrin for the first time.

A lone break of the Australian's serve in the second was enough for the 20-time Grand Slam winner to seal the win and set up a showdown with fifth seed Alexander Zverev.

Dominic Thiem is playing his first tournament since mid-March
Dominic Thiem is playing his first tournament since mid-March AFP / GABRIEL BOUYS

"It was difficult. The court today was slippery, it was very fast, very dry conditions. The ball was flying a lot. The beginning of the match was super tough," said Nadal who wrapped up a 12th Barcelona title last weekend.

"It was an important victory, he came here playing well."

Zverev, the 2018 champion in Madrid, defeated Britain's Dan Evans, who upset world number one Novak Djokovic at Monte Carlo last month, 6-3, 7-6 (7/3).

Nadal has a 5-2 career lead over Zverev, winning all three meetings on clay.

Tsitsipas, who held a championship point against Nadal in Barcelona last week, suffered a 7-6 (7/4), 6-4 loss to Norway's Casper Ruud.

Ashleigh Barty is one win away from a fourth title of the season
Ashleigh Barty is one win away from a fourth title of the season AFP / GABRIEL BOUYS

The 22nd-ranked Ruud is through to his third Masters 1000 quarter-final in a row on clay after earning his first win over a top-five player.

Dominic Thiem, the Madrid runner-up in 2017 and 2018, also reached the last eight after beating Alex de Minaur 7-6 (9/7), 6-4.

US Open champion Thiem, playing his first tournament since Dubai in March, will next meet John Isner following the American's 7-6 (7/4), 3-6, 7-6 (7/4) victory over Andrey Rublev.

Sixth seed Rublev had made at least the quarter-finals in all seven of his tour-level appearances this year before arriving in Madrid.

He had also stunned Nadal in Monte Carlo.

Rafael Nadal is bidding for a sixth Madrid Open title
Rafael Nadal is bidding for a sixth Madrid Open title AFP / GABRIEL BOUYS

Medvedev exited following a 6-4, 6-7 (2/7), 6-1 defeat by 16th seed Cristian Garin of Chile.

The Russian had won his first match in Madrid in his third appearance on Wednesday, but the Russian was sent packing by an opponent far more at ease on clay.

The two-time Grand Slam runner-up heads to Rome next.

He has never won a match in the Italian capital, and he has failed to get past the first round of the French Open in four previous tries.

Alexander Bublik of Kazakhstan progressed with a 6-4, 6-3 win over in-form Russian Aslan Karatsev.

Barty cruised into the women's final with a 6-4, 6-3 victory over Badosa, as the world number one avenged last month's loss to the Spaniard in Charleston.

Barty, the 2019 French Open champion, is targeting a fourth title of the year following her wins at Melbourne's Yarra Valley Classic, in Miami and on clay in Stuttgart.

The top seed shaded a tight opening set against the 62nd-ranked Badosa, the first Spanish woman in the tournament's 12-year history to make the semi-finals.

She converted her third set point when Badosa doubled faulted at 4-5, a single break of serve proving decisive for the Australian.

Barty dropped serve herself to start the second set but broke twice in succession to regain control and make it 16 straight wins on red clay.

"You have to learn from every game, every match you play against an opponent," said Barty.

"I definitely learnt a lot from the match we played in Charleston. There was a small adjustment."

In Saturday's final, she will play fifth seed Aryna Sabalenka of Belarus who eased past Russia's Anastasia Pavlyuchenkova 6-2, 6-3.