The plane is kept away from the terminal as police in masks keep watch and some passengers are taken to hospital.
Dozens of vehicles surrounded the aircraft at JFK airport, with people on board reportedly coughing and feverish.
Experts from the Centers for Disease Control and Prevention (CDC) were called to the scene.
The plane - a double-decker Airbus A380 - arrived in New York from Dubai shortly after 9am and was kept away from the terminal.
An airline statement said: "Emirates can confirm that about 10 passengers on #EK203 from Dubai to New York were taken ill.
"On arrival, as a precaution, they were attended to by local health authorities... The safety & care of our customers is our first priority."
An earlier statement from the CDC said around 100 people, including some crew, had complained of feeling ill.
Eric Phillips, a spokesman for New York's mayor, said 521 people had been on board and that 10 sick passengers were being taken to Jamaica Hospital.
Mr Phillips later tweeted: "All the passengers are off and have been evaluated. 19 sick. 10 to hospital and 9 refused medical attention.
"Health officials are processing tests now to determine the cause. Symptoms still pointing to the flu."
A passenger on board, Larry Coben, said people were allowed off after filling out paperwork from the Centers for Disease Control and Prevention.
His pictures showed people having their temperatures taken with a laser device as they disembarked, and a police officer wearing a mask.