He was charged with first-degree sexual abuse, a felony, and endangering the welfare of a child, a misdemeanour.
A preliminary felony hearing in the case was scheduled for Monday. Hussain appeared in court, but the hearing did not take place.
A preliminary hearing is usually held when a person is in jail. It has to be held within 144 hours of the time the person was arrested, or he or she can be released.
With an interpreter translating for Hussain, who doesn't speak much English, Essex County Assistant District Attorney Jamie Martineau asked the judge to reschedule the hearing, continue to keep Hussain on bail, not return his passport and require him to stay in the county until the matter is resolved.
Martineau said Hussain is a flight risk, noting he had a flight booked to leave the country from New York City on Saturday, the day after he was bailed out of jail.
"He is not going to leave the country," Barrett said. "He absolutely denies these charges. We've been presented a plea offer where essentially he could accept a minor charge and walk out of here today. But he is electing not to do that to stay in the United States and fight these charges."
As a potential compromise, Barrett suggested the judge could keep Hussain's passport and allow him to stay in the state, including New York City, where Barrett said there are people with whom Hussain can stay.
Barrett said the plea deal Hussain rejected would have been an admission of guilt to endangering the welfare of a child, with a conditional discharge.
Since Hussain's arrest, some people have questioned the girl's account, including Abid Khan, Hussain's traveling companion who returned home to India. Last week, Khan said the girl had followed him and Mr Hussain around in an affectionate way during their stay in Saranac Lake, the report said.
One day when they returned to the Porcupine Inn on Park Avenue, where they were staying, Khan said they found the girl playing pool with other guests.
Mr Khan said Hussain told him the girl tried to make an advance on him, but he turned her away.