Police were called to Queen’s Park in Chorley New Road, Bolton, at 10.20am on Sunday after reports that six dogs were dangerously out of control.
Emergency services found a 60-year-old woman who had been bitten on the hand and a man with an injured leg.
Greater Manchester Police said: “Armed response officers were deployed to the scene and two dogs were humanely destroyed and two dogs have been seized.
“Officers are currently searching the area for another dog who is believed to be part of the group.”
A 38-year-old man was arrested on suspicion of affray and allowing a dog to be dangerously out of control in public.
Superintendent Chris Allsop said: “Highly trained officers responded quickly to the scene and took control to avoid the situation from escalating.
“The officers acted with our number one priority in mind; to protect people and prevent them from harm.
“I’d like to thank to the public for their patience and full cooperation as we remain at the scene and carry out our inquiries.”
Two roads that were closed in the area have now reopened, and officers are trying to find the injured dog.
Anyone who sees a dog loose in the area is asked to tell police, while witnesses or those with information are urged to call police on 101 or Crimestoppers on 0800 555 111.