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Svitolina then told him to do the same on the men's side against Matteo Berrettini Wednesday.
"Now he must step up his game," Svitolina quipped in his direction in the on-court interview.
"We are definitely pushing each other to the quarter-finals and now the semis."
Svitolina and Monfils share a popular Instagram account, where they can be seen on holiday and kissing on the practice court.
The two play on alternating days in New York, so they can watch each other without disturbing their practice routines, though Svitolina has talked at length about the benefits of training together at times.
"He's hitting pretty strong and very heavy. So I think definitely it helps me to practise with him," she said Tuesday.
"But I need to find a balance between his shots because, if I only practise with him, I'm going to be only in defence most of the time, and that's not so good, either.
"So I have to find balance when I'm training with him. I also train with my coach to do the attacking shots as well. It helps but I have to adjust as well."
The 24-year-old said that the pair "enjoy each other completely on the court as well" as she advanced into her second straight grand-slam semi, following her last-four appearance at Wimbledon two months ago. She is assured of a return to the world-number-three ranking next Monday.
The winner of 13 tournaments, a highlight of those the 2018 season-ending tour championships, would even climb to a career-high second in the world if she wins her maiden major trophy Saturday.
Monfils, 33, made the 2008 French Open and 2016 US Open semis, has eight career titles and a best ranking of sixth in the world from three years ago.
Monfils delights fans with his charisma and great shots, such as a 360-degree turn-and-smash at the net on match point in the second round against Marius Copil.
"Yeah, he always hits that. That's normal for me. That was like a normal day in the office," Svitolina said, adding that she has also had a go at it.
"I have been practising as well. Actually this morning I did it. Not as good, though. I turn and then I hit."
Whether she would also show it Thursday in the semi, against either the mighty Serena Williams or Wang Qiang, she did not reveal.
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