The dog walker is described as a white female, aged 30-35, tall, of slim build with blonde hair. Her dog is described as a non-castrated, brindled coloured Staffordshire type with a white left paw and ankle.
She was reported as walking along by the trees near a London Road entrance to the park at around 10:30am. Both dogs were off the lead.
Police in Sutton say this is the second incident reported to them involving a brindle coloured Staffordshire type dog in the same park.
PC Heath Keogh, who set up the borough's LEAD initiative in Sutton to encourage responsible dog ownership, said dog on dog attacks are not recordable offences even if a dog is attacked or injured.
However, police have a range of powers to deal with anti-social behaviour by dog owners to make sure dogs are kept under control including engaging with owners and the issuing of warning letters, control orders and community protection notices.
PC Keogh explained that a dog owner is responsible for their dog's action, not their dog.
"When it comes to a member of the public feeling threatened by a dog, a person will only have to have a reasonable apprehension that a dog will injure or attack them for an offence to have been committed under the Dangerous Dogs Act.
"We would like this dog owner to come forward so we can engage with her and give appropriate advice," he said.
Police in Sutton are liaising closely with their colleagues in Surrey as much of Nonsuch Park falls within Surrey's policing area.
The dog owner is being asked to contact Sutton Police Station on 101 or Sutton's Safer Parks Police Teams on 020 8721 2268.