Japan's Osaka, 24, who has won four Grand Slam titles, lost for the sixth time in eight matches, going down 6-4 7-5 to China's Zhang Shuai.

The US Open starts on 29 August.

There will be concern for both Gauff and Osaka as they head from Cincinnati to Flushing Meadows in New York for the final major of 2022.

American Gauff has risen to 11th in the world since finishing runner-up to Iga Swiatek at the French Open in June.

She also claimed top spot in the women's doubles rankings after winning the Canadian Open title on Sunday.

Gauff would surely have been eying a long run in both the singles and doubles at the US Open.

But she suffered the injury to her left ankle in the 11th game as she went 6-5 down, before Bouzkova served out for the first set. Gauff then retired hurt when she was broken again in the opening game of the second set.

Former British number one Greg Rusedski, speaking as part of Amazon Prime's coverage of the Cincinnati tournament, said: "Let's hope the ankle is not too serious because she has played some great tennis to reach number one in doubles.

"You would have expected her to come through this match and hopefully she will be OK for the US Open."

Osaka's immediate concerns will be around her worrying slump in form after falling to 39th in the world on the back of a series of disappointing results.

Since finishing runner-up to world number one Swiatek at the Miami Open in April, the double US Open and Australian Open champion has failed to make it past the second round in five WTA events, losing in the first round three times, including at the French Open.

Osaka has also had injury issues, only recently returning from an Achilles problem that forced her to pull out of Wimbledon.

She also retired from her first round match at the Canadian Open last week, complaining of pain in her lower back.

'Maturing' Kyrgios marches on

Nick Kyrgios has won nine of his 12 matches in ATP 1000 Masters events this year

Nick Kyrgios' US Open prospects look much better as the Australian beat Spain's Alejandro Davidovich Fokina 7-5 6-4 to set up a second round match against American Taylor Fritz in Cincinnati.

Since his march to this year's Wimbledon final, where he lost to Novak Djokovic, the 27-year-old has won the seventh ATP title of his career in Washington and reached the quarter-finals of the Canadian Open.

He needed just one break of serve to take the opening set against Davidovich Fokina and looked to be heading for a regulation victory when breaking the Spaniard in the opening game of the second set.

His opponent, though, broke back to level at 2-2, prompting Kyrgios to fire a ball into the stands and pick up a warning for a code violation.

But the world number 28 held his next service game to love and broke for a second decisive time to take the match.

Despite one brief outburst, Kyrgios played the rest of the match in good spirits and explained the secret of his positive approach.

Speaking to Amazon Prime after the match, he said: "My consistency is a lot better now than it has been in the rest of my career.

"My motivation is the secret, I am playing for a lot more than just myself these days.

"I am 27 now so it's about time I started maturing, but I am just having a lot of fun and making the most of it while I'm out here [in the United States]."

World number one Daniil Medvedev is also through to the third round after beating Dutchman Botic van de Zandschulp 6-4 7-5.

The 26-year-old US Open champion started strongly against Van de Zandschulp with an immediate break before going on to win the first set.

In the second set, Medvedev had to save a set point but several double faults by Van de Zandschulp allowed the Russian to capitalise before sealing victory.

"Two breaks on these fast courts is enough to win and I should have done a little bit better on my serve," said Medvedev.

"But the first match is tricky and I'm happy that I managed to go through, and hopefully I will raise my level in the next matches."

Medvedev will face either American Tommy Paul or Canada's Denis Shapovalov next.