Roger Federer looks bang on course to launch his comeback next month after the 20-time grand slam winner posted an encouraging video of a practice session on Sunday.

The Swiss turned 41 on August 8 and has not played a match since losing to Hubert Hurkacz in the quarter-finals at Wimbledon last year.

He has undergone knee surgery in the meantime, his third such operation inside two years, but is determined to resume his playing career and looked to be enjoying being back on court in a short clip posted to Instagram.

Federer is due to play in the Laver Cup, which will be staged in London from September 23 to 25, with Rafael Nadal, Andy Murray and Novak Djokovic set to join him in an illustrious Team Europe line-up.

It will be a reunion for the 'Big Four', who were once so dominant as a group, with Stefanos Tsitsipas and Casper Ruud also on the European team, which will be captained by Bjorn Borg.

Federer is then entered for the Swiss Indoors in his home city of Basel, where he has been champion 10 times previously and holds a remarkable 75-9 win-loss career record.

It remains to be seen whether that October 24-30 tournament is a farewell appearance, or another step towards playing regularly on the ATP tour once more.

At Wimbledon in July, Federer appeared at a parade of former champions and said he wished to play at the All England Club again, which points to him planning to play in the 2023 season.

"I hope I can come back here one more time," said Federer, an eight-time Wimbledon champion.

Petra Kvitova fended off home favourite Madison Keys to reach the Western and Southern Open final and secure a return to the world's top 20.

The two-time Wimbledon champion battled to a 6-7 (6-8) 6-4 6-3 victory in two hours and 18 minutes, saying the match had been "mentally so tough".

Both players had taken significant scalps to reach this stage, with Keys knocking out world number one Iga Swiatek and Kvitova ousting Wimbledon runner-up Ons Jabeur.

Kvitova reached semi-finals at Cincinnati in 2012 and 2018 and lost three-set battles on both occasions, but this time she came good in the decider to earn a first final appearance at the event and a 40th overall in her WTA career.

The 32-year-old Czech slipped an early break adrift in the opening set before fighting back to force a tie-break. She came from behind in that, too, to level up at 6-6, only for hard-hitting American Keys to win the next two points, the first with a sizzling backhand winner.

Kvitova sensed she was still firmly in the match, however, and surges to 3-1 in each of the next two sets provided the platform to fully turn around the contest.

A 12th WTA 1000-level final awaits her, and she will go to 20th in the rankings should she lose Sunday's final and up to 15th if she carries off the title.

Kvitova said of her win: "It was incredibly tough. I expected it would be tough, but I didn't expect it that tough. Madison played unbelievable.

"She played an amazing tie-break, but I tried to keep going, be there and be a little calmer than before and wait for the chances. It was an incredible match, and I'm really happy that I made it somehow.

"I don't really care about the ranking. Being in the final, that's more important, and that's why I'm playing tennis: to be in finals and fighting for a trophy."

She awaited the winner of the second semi-final between Aryna Sabalenka and Caroline Garcia.

Madison Keys stunned top seed Iga Swiatek 6-3 6-4 to book her place in the quarter-finals of the Western & Southern Open on Thursday.

Keys arrived in the contest seeking a first career victory over a reigning world number one in six attempts, having failed to win a set in each of the previous five.

The 2019 champion had also lost her previous two showdowns with Swiatek, including a resounding 6-1 6-0 defeat in the Indian Wells Open quarter-finals earlier this year.

After the first games went with serve, Keys went into overdrive; reeling off nine successive games to take full command of the contest at 6-3 5-0.

Swiatek rallied as she rescued match point to avoid the bagel, winning 11 of 12 points to reduce the second-set deficit to 5-3.

The winner of six WTA titles this season, the Pole saved a further match point in the ninth game before claiming what appeared to be another crucial break of serve.

However, former US Open finalist Keys responded magnificently to make it third time lucky against Swiatek, who is now 4-4 since her 37-match winning streak ended with defeat by Alize Cornet at Wimbledon.

The prize pot for this year's US Open will top $60million for the first time in the competition's history, event organisers have announced.

The 2022 edition at Flushing Meadows gets under way on August 29 and runs for just under two weeks.

It was revealed on Thursday that $60m will be up for grabs, topping the previous record of $57.5m from last year, with both singles champions to receive $2.6m.

Players will be given $80,000 for making the main draw and $121,000 should they make it to the second round. Runners-up in the singles will pocket $1.3m.

In the doubles, the champions will receive $688,000, the runners-up $344,000 and the semi-finalists $172,000.

Emma Raducanu admitted she has been "too tense" in recent defeats and is aiming to play more freely as she prepares to defend her US Open title.

Raducanu has struggled to follow up last year's shock win at Flushing Meadow, but she appears to have hit form at just the right time.

The 19-year-old has beaten both Serena Williams and Victoria Azarenka in straight sets during this week's Western and Southern Open in Cincinnati, dropping just four games against Williams and two against Azarenka.

Her win against Azarenka on Wednesday was just Raducanu's third career victory against a top-25 opponent, after beating Belinda Bencic and Maria Sakkari on her way to winning last year's US Open.

Speaking after her win against the Belarusian, Raducanu said: "I think I have tried a lot of things, and this year I think some matches, I have lost a lot of matches from leading situations and probably just played too tense.

"I think that I just need to swing, and I just said that these two tournaments, or this tournament especially, I'm just going to swing freely and take that and see what happens."

Raducanu also said she will wait until after the tournament to reflect on beating two multiple grand slam champions with relative ease.

"I think after the tournament I actually have to take a step back and be like, 'Look, you just beat two unbelievable champions'," she said. "It's a pretty good achievement. You know how difficult the wins are to come by."

The number 10 seed will face seventh seed Jessica Pegula in the third round in Cincinnati on Thursday.

Taylor Fritz believes there would be no "harm" in Novak Djokovic competing at the US Open, although the American admits he is conflicted about whether he should be allowed.

Wimbledon champion Djokovic has his route to Flushing Meadows blocked by red tape at present, with the United States refusing to allow unvaccinated foreign visitors to enter the country.

US tennis authorities have pledged to adhere to government rulings surrounding COVID-19 protocols, despite including Djokovic on their entry list.

For Djokovic to be allowed into the United States, it appears he would require a change of policy from law-makers, or he would have to be considered suitable for an exemption.

Among those who would be eligible for such an exemption are "persons whose entry would be in the national interest".

With 21 grand slam titles, Djokovic sits second on the men's all-time list behind Rafael Nadal, who has 22 victories at the majors. 

And Fritz, ranked 13th in the world, said that while the situation is complicated, there is part of him that thinks Djokovic should be allowed to take part in the competition that starts in under a fortnight.

"It's tough," he said. "I think on one side of it, I think it's tough to make certain exceptions to the rules for certain people.

"I don't know how I feel about that, but then, at the same time, we're not the most COVID-safe country in general with how we are doing things.

"So it does seem like, what's the harm of letting the best player in the world come play the US Open?

"But like I said, at the same time, it's conflicting, because I don't know how I feel about making special exceptions just for one person because of who they are.

"So I see both sides of the argument, to be honest. It's tough to differentiate, obviously. It's good for every player if Novak is not in the draw."