Britain’s Jodie Burrage and Jay Clarke were among the small group of qualifiers who managed to step on to court earlier.
Burrage, who reached the second round proper in 2023, trailed 5-3 against Dutch opponent Arantxa Rus when they were forced off after 41 minutes of play.
Clarke was also a break down by the scoreline against French top seed Arthur Cazaux.
Fellow Britons Billy Harris, Francesca Jones, Harriet Dart and Oliver Crawford did not even get out on court.
Rain is forecast for the rest of Wednesday before clearing up on Thursday.
There is still plenty of time for the three rounds of qualifying to be concluded, with the singles main draws starting on Sunday.
The US Open is the final Grand Slam event of the 2025 season.
