There will be no Andy Murray, who retired from the sport after his Olympic men’s doubles quarter-final defeat in August.
British number two Cameron Norrie is also missing with a forearm injury.
Jack Draper, the British number one, will be seeded for the US Open, with Dan Evans also in the main draw.
They both safely made it through the first round, as did British women’s number one Katie Boulter and Harriet Dart.
They were joined by Jan Choinski, who came through qualifying before losing his opening match in five tight sets.
Emma Raducanu, champion in 2021, also lost in first round to fellow Grand Slam winner Sofia Kenin.
In the men’s doubles, Joe Salisbury and American partner Rajeev Ram are the three-time defending champions.
