Despite a straightforward fourth-round win over Diana Shnaider, something had irritated Jessica Pegula.

“The most annoying thing is that people think I have a butler,” the 30-year-old American said on Monday.

“That I get chauffeured around. That I have a private limo. That I fly on a private jet everywhere.

“I am definitely not like that.”

The misconceptions about her lifestyle come from her family’s wealth. According to Forbes magazine, her father Terry, an oil and gas tycoon, is worth $7.7bn (£5.9bn), making him the 371st richest person in the world.

She is not the only billionaire’s daughter in the last four of the US Open on Thursday.

On the other side of the draw is Emma Navarro. Her father Ben owns a credit card empire and, according to Forbes, is worth $1.5bn ( £1.1bn).

They are numbers that put even the $3.6m (£2.7m) prize money for the Flushing Meadows champion – the biggest cash prize in tennis – in the shade.

But there is some way to go before Pegula and Navarro can book in for a box-office, big-money, all-American battle for the title.

Two-time Australian Open champion Aryna Sabalenka, the highest seed remaining in the draw, has shown strong form en route to her meeting with Navarro, while outsider Karolina Muchova has a happy habit of claiming big scalps as she prepares to face Pegula.

BBC Sport looks at the form and the stories of the four semi-finalists.

Head-to-head: 1-1; last meeting: French Open fourth round – Sabalenka won 6-2 6-3.

Even with world number one Iga Swiatek in the draw, Sabalenka’s hard-hitting game has made her the one to beat on the hard courts.

The Belarusian world number two retained her Australian Open title in January in emphatic fashion, storming to the trophy without dropping a set.

However, in March her former boyfriend, Konstantin Koltsov, died in what she called “an unthinkable tragedy”.

Sabalenka continued playing – something she said, with hindsight, she should not have done – before she struggled with a stomach problem in her French Open quarter-final loss and a shoulder injury ruled her out of Wimbledon.

However, she claimed her first title since Melbourne in Cincinnati in the build-up to the US Open, and the fast New York courts suit her huge serve and devastating forehand. She was runner-up in last year’s tournament, going down to Coco Gauff.

Sabalenka on Navarro: “Beautiful to see – she’s working hard, playing really great tennis, smart tennis, moving well, hitting pretty heavy shots.”

Navarro beat second seed Sabalenka on the Indian Wells hard courts in March for the biggest win of her career by ranking.

She has been one of the stars of the year, winning her first WTA title in Hobart in January before consistently improving on her Grand Slam showings.

She reached the Australian Open third round, the fourth round at Roland Garros and the quarter-finals at Wimbledon.

She has also had some statement wins, including beating Coco Gauff at Wimbledon and the US Open, ending her compatriot’s title defence at the latter.

Navarro is a highly intelligent player who use slices and drop shots to mix up the pace of rallies.

Navarro on her mentality: “I always step on the court believing that I have a chance to win, and it’s always my priority to just put my best foot forward.”

Head-to-head: Pegula 1-0 Muchova; last meeting: Cincinnati Open second round – Pegula won 5-7 6-4 6-2.

It is a long-awaited first Grand Slam semi-final for Pegula, who lost her previous six quarter-finals.

She split with long-term coach David Witt after the Australian Open, where she lost in the second round, before her clay season was interrupted by neck and back injuries.

A second-round Wimbledon exit was followed up by a superb run, with Pegula defending her Canadian Open title before reaching the Cincinnati final a week later.

She has reaching the US Open last four without dropping a set.

Born in Buffalo, New York, Pegula’s parents own NFL side the Buffalo Bills. They supported her through her junior years as she struggled with injuries.

She claimed a stunning win over Swiatek in the quarter-finals, barely letting up in intensity and returning serve solidly to end her last-four hoodoo.

Pegula on Muchova: “She’s so good, so talented, so athletic. I love how she just doesn’t play and comes back and beats everybody. I’m gonna have to bring my best tennis.”

Muchova has played only six tournaments since last year’s US Open, having returned to the WTA Tour in June following wrist surgery.

Despite a lack of playing time, Muchova has yet to drop a set in New York and has seen off some big names.

She beat four-time Grand Slam champion Naomi Osaka in the second round and fifth seed Jasmine Paolini in the fourth, before overcoming an apparent stomach illness in her quarter-final win over Brazil’s 22nd seed Beatriz Haddad Maia.

Czech Muchova was runner-up to Swiatek at the French Open in 2023 and is the only player to have taken a set off the world number one in a Roland Garros final in four attempts.

A crafty player, Muchova has been compared to Justine Henin and Roger Federer for her adeptness around the court.

Muchova on her wrist surgery: “This last injury was one of the worst ones that I had. Looking back, I’m like, oh, it actually flew by and I feel strong again.”