At around 1.30pm a shot was heard from inside the address.
When Police entered the property they found the man alone with a serious injury.
Lynch said police had not fired any shots during the operation. The man is in a critical condition at Middlemore Hospital.
Police said the children were known to the man but he not believed to be their father.
Lynch said police will remain at the scene as officers look to build a fuller picture of the circumstances.
"All cordons have now been lifted and Police would like to thank the neighbouring community for their patience during this time."
The operation had brought the area to a standstill this morning.
Sutton Cres and Rollerson St were no-go zones as police asked people to stay away.
Several residents on the nearby Sheehan Avenue sat outside their properties watching the incident unfold.
A resident told the Herald armed police cordoned off the area earlier in morning. Police cars were stationed at either end of the normally busy Rollerson St.
Wilson Superette worker Amit Singh said when he arrived for work at 6.45am he was initially barred by police from entering the area.
He waited two hours before he managed to use a pathway get to work.