ROME -- When Paula Badosa was away from tennis for six months with a stress fracture in her back, Coco Gauff was one of the players who reached out.

“She was texting, checking in,” Badosa said a few days ago here at the Internazionali BNL D’Italia. “I love when there’s great athletes -- and a nicer person. Always wish these kind of players the best.”

Except, perhaps, when they’re on the other side of the net.

“I always root for her,” Gauff said. “She was No. 2 in the world, playing well. She’s always dangerous regardless of how the past weeks have gone. I think she’s getting back to her top form.”

On Monday, Gauff won a tight three-setter, 5-7, 6-4, 6-1.

On Tuesday, she will meet No.7 seed Zheng Qinwen in a quarterfinal match. 

They are the two youngest players among the Hologic WTA Tour’s Top 25, but they’ve never played. Going forward, you get the idea they’ll be seeing a lot of each other.

Gauff and Zheng actually met as juniors in the 2018 Orange Bowl. Gauff, then only 14, was a winner in three sets. And wouldn’t you know it -- six years later, she had the score right.

“I remember being down a break in the third set and coming back,” Gauff said. “I think I won 6-4 or something like that.”

Indeed, it was 6-1, 3-6, 6-4.

“Obviously we’re both different players,” Gauff added. “We’ve both done a lot since then. It should be a fun matchup for the next round.”

Gauff is ranked No.3 and Badosa is still outside the Top 100, but this was a battle. Badosa had won three of their four previous matches – including the most recent last year in Madrid. Gauff won a total of only three games.

This time, it was a vastly different outcome.

Gauff shrugged off that first set defeat and went on to break Badosa, a quarterfinalist here last year, seven times. Gauff is now 6-2 on clay for 2024.

Badosa could become the first Spanish player to make consecutive quarterfinals at the Internazionali d’Italia since Garbiñe Muguruza in 2016-2017.

Coming in, Gauff didn’t have a specific goal as far as advancing. She was more concerned with getting her serve right. She lost to Madison Keys in three sets recently in the fourth round at Madrid and 13 double faults were one of the leading culprits.

“I feel like I’m returning well, hitting off the ground well,” Gauff told reporters. “I think I just fix that detail, it will save me, and maybe those matches would turn into winning in straight sets instead of losing in three sets.

“I feel like that’s the part that if I can work that through, I think it will set me up for a very good Roland Garros.”

Gauff won 76 percent of her first-serve points.

“I’ve been working on it,” Gauff said of her serve. “I think today, even though the double-faults were there [11], I think it was still a better performance than it was last night. For me, the focus will be getting more first serves in.

“I think it’s just finding the balance of going big but also knowing when to slow down the pace just to get the serve in. Honestly, I bet on myself to continue to go big. I know when I go big and my serve goes in, it’s dangerous.”

Badosa broke through in 2022, winning the title in Sydney and reaching the semifinals at Indian Wells, Stuttgart and San Jose. A year later, the Spaniard reached the quarterfinals here in Rome but withdrew from Roland Garros. She tried to come back two months later at Wimbledon but retired from her second-round match, which began her extensive leave.

Doctors discovered a serious back injury, the extent of which wasn’t fully understood publicly until very recently. For anyone, the back is a critical part of the core -- but for professional tennis players, it’s the part of the body that absorbs the stress of those torquing groundstrokes. Doctors told Badosa it would be “complicated” to continue her career.

At the age of 26, she’s still dealing with daily chronic pain.

"For me,” Badosa said on the WTA Insider Podcast, “being able to play three or four more years would be amazing.”

She was a modest 6-9 for the year coming into Rome but put together some nice wins over Mirra Andreeva, No.21 seed Emma Navarro and Diana Shnaider. The last two matches went three sets and fatigue seemed to be a factor in the latter stages of her match against Gauff.

Gauff won 30 of 46 points in the final frame.

Gauff has been on tour so long, it’s easy to forget she turned 20 in March. Every match, it seems, produces another milestone. She’s now even with Caroline Wozniacki with an impressive total of 60 match-wins in WTA 1000 events before the age of 21. She’s also only the fourth American to win 10 matches at the Italian Open before turning 21, joining Chris Evert and the Williams sisters.