The 36-year-old is the star attraction on Arthur Ashe Stadium which on Friday bid an emotional farewell to Serena Williams, who ended her career on 23 majors, one more than the 22 that Nadal currently possesses.
World No 3 Nadal and Gasquet were born just 15 days apart but their careers have enjoyed different trajectories.
As well as losing 17 times in 17 meetings with Nadal on the professional circuit, the Frenchman has dropped the last 31 sets against his childhood rival.
“We are both 36 years old, but in terms of recovery, we are not equal,” said Gasquet.
Nadal arrived at the US Open having played just one match since forfeiting his Wimbledon semifinal due to an abdominal injury.
In New York, he has dropped the opening set in both of his first two matches.
Against Fabio Fognini, he even injured himself with his own racquet which bounced off the court and crashed into his nose.
Nadal also has concerns over his wife Mery Perello – due to give birth to the couple’s first child in the coming weeks.
Spanish media reports claimed she had been admitted to hospital as a precautionary measure.
“My wife is fine, but in the end these are situations that are sometimes more difficult to manage when you are away from home,” said Nadal.
British seventh seed Cameron Norrie made the fourth round for the first time with a comfortable 7-5, 6-4, 6-1 win over Danish teenager Holger Rune.
In the women’s draw, 21st seed Petra Kvitova downed ninth-seeded Garbine Muguruza after saving two match points.
Kvitova came through 5-7, 6-3, 7-6 (12/10) and will take on Jessica Pegula of the USA for a quarterfinal place.
Two-time Wimbledon champion Kvitova said she had drawn inspiration from Serena Williams when the American legend saved five match points before eventually being eliminated in her third round tie.
“I watched Serena and it was amazing how she was saving match points, so I tried to do the same today,” said two-time US Open quarterfinalist Kvitova.
Muguruza’s defeat, her sixth in seven meetings against the Czech, left only four of the top 10 women’s seeds in the tournament.
One of those is eighth-seeded Pegula who ended the run of Chinese qualifier Yuan Yue, 6-2, 6-7 (6/8), 6-0.
Later Saturday, world No 1 and French Open champion Iga Swiatek seeks a second successive fourth round spot when she faces unseeded Lauren Davis of the USA.
Swiatek has never lost to a player outside the top 100 at the Slams.
Sixth seed Aryna Sabalenka, who made the semifinals in 2021, saved two match points in her second round win over Kaia Kanepi.
The Belarusian faces French qualifier Clara Burel, the 131st-ranked player who knocked out Wimbledon champion Elena Rybakina in the first round.
Chinese teenager Zheng Qinwen plays Germany’s Jule Niemeier. Big-serving Zheng has hit a tournament-leading 32 aces so far, one more than the now retired Williams.