The 5:30 a.m. collision, at an intersection in Greenwich Village, sent the bus over the sidewalk, through scaffolding, and into the side of a bank building, the New York Post reported.
The driver of the box truck, who reportedly suffered only minor neck injuries, was taken to the hospital in police custody.
Transit officials said the 50-year-old bus driver died at the scene, the New York Daily News reported.
They said police were chasing the stolen truck at the time of the collision.
Three other people were hospitalized with minor injuries, police said.
One of the victims was a street vendor whose food cart was in the path of the collision, said Kevin Ortiz, a spokesman for the Metropolitan Transit Authority.