Staffordshire Police said officers were called to a property on Corporation Street at about 7.30am on Sunday following reports from the West Midlands Ambulance Service.
Two children were pronounced dead at the scene and have since been named as two-year-old Meraj Ul Zahra and three-year-old Abdul Momin Alfaateh.
Their next of kin have been informed and are being supported by specially trained officers.
Police confirmed that a 43-year-old woman from the Stafford area has been arrested on suspicion of murder and remains in custody for questioning.
Detective Inspector Kirsty Oldfield said: “We are working hard to understand what happened leading up to these two children tragically losing their lives.
“We ask that people do not speculate at this stage as it is distressing for family and friends and could hinder our inquiries.
“We understand this incident may cause concern in the local community, but we do not believe there is a wider threat to the public at this time.”
A police cordon remains in place at the property while investigations continue.
Anyone with information that could assist the investigation is urged to contact Staffordshire Police on 101 or via the live chat service on their website, quoting incident number 147 of October 12. Information can also be given anonymously through Crimestoppers on 0800 555 111.
