The 27-year-old, who was arrested on suspicion of child neglect, will return to a south London police station in mid-January.
The investigation, which is being led by detectives from the South Area Command Unit, is continuing.
Four young children died following the fire at a house in Collingwood Road on the evening of Thursday, 16 December.
Firefighters, paramedics and police officers responded to the incident shortly after 7pm
The four children were rescued from the building but despite the best efforts of the emergency services, they died a short time later.
Although formal identification is yet to take place, officers are confident they are twins Leyton and Logan Hoath, aged three and Kyson and Bryson Hoath, aged four. The boys were all brothers.
Chief Superintendent Dave Stringer, policing commander for Sutton, Croydon and Bromley, said: “The thoughts of everyone at the Metropolitan Police are with the family and friends of the four young boys who lost their lives in this tragic incident.
“We know it will have come as a great shock to the local community and as part of that community, we are here to offer whatever support we can.
“Local detectives continue to investigate and will be working with colleagues from the London Fire Brigade to determine what caused the fire and to investigate any other relevant matters.
“We will continue to work with the local authority and the wider community to support local residents as they come to terms with this tragic loss.”