NEW YORK — With just a few days remaining until a key deadline, more than 32,000 people have applied to the federal compensation fund for people with illnesses that might be related to toxic fallout from the Sept. 11 attacks, program officials said.

Congress has authorized paying as much as $2.78 billion to people exposed to the tons of thick dust that fell on Manhattan when the World Trade Center collapsed.

The first deadline to apply for a payment comes Thursday, and as that date has approached, the number of applicants has soared. As recently as late June, only 19,733 people had applied. More than 6,000 registrations have been completed just since Sept. 15.

Applications have come in the greatest numbers from firefighters, police officers and construction workers who spent months on the smoking debris pile.

The Oct. 3 deadline applies to anyone who was ill with a condition covered by the fund when it began operating in 2011. That group includes people with several types of respiratory and digestive system ailments.