It happened at 31st Street and Hoyt Avenue in Astoria just before 6 a.m.
Authorities said one of the vehicles may have attempted to cut off the other, causing the crash.
The bus had no children on board, just the driver and a matron.
No serious injuries were reported.
The vehicles crashed on the local streets, just off the ramp from the RFK Triborough Bridge.
Authorities say the driver of the allegedly stolen vehicle was taken into custody.
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