A 37-year-old man and a 34-year-old woman have been arrested on suspicion of murder in connection with the incident, police said.
Police were called to the property at around 7.30am on Friday after locals expressed concern for the welfare of those inside.
Yorkshire Ambulance Service and Yorkshire Air Ambulance raced to the scene and transported six children to hospital.
A spokesman for South Yorkshire Police said there would a large number of officers in the area throughout the day as inquiries continued.
The incident took place close to the Hartley Brook primary school and witnesses reported seeing an air ambulance land in the grounds of the school.
Local residents claimed they had seen two children being taken out of the house by paramedics before being rushed away in an ambulance.
There were initial reports that the incident involved firearms, but South Yorkshire police denied that there had been a shooting.
Detective Chief Inspector Simon Palmer said: "Detectives and specialist officers remain in the Gregg House Road area carrying out inquiries and we are working hard to ascertain the exact circumstances surrounding this incident.
"We will remain in the area this afternoon and into the evening carrying out inquiries and providing reassurance.
"I appreciate the events of this morning have caused concern locally, but Id like to reassure you that there is no wider risk to the community.
"To allow the investigation to progress, I would please ask that people are mindful of what they are posting on social media, and the potential distress this may cause."
It is understood the youngsters were taken to The Sheffield Childrens Hospital in the Broomhill area of the city.
Jonathan Campbell,50, who lives a few doors up from the house where the incident took place, spoke of his shock.
He said: "The police knocked on our doorstep and told us there had been a domestic. It was quite a shock to see so many police cars and ambulances outside. I saw a baby of about six months old being carried to an ambulance by a paramedic.
Mick Whittaker, 61, milkman who has lived on the same street for 11 years, said: "I pulled up from my round at about 8am and there were sirens wailing down the street. Twenty minutes later the air ambulance landed in Hartley Brook school.
"I didnt know the family and asking around it seems nobody did. There have been sirens down that end of the street before but this is horrific."
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