The blaze broke out at the Taufiq family home in Wood Hill, half a mile from the scene of an earlier stabbing.
Three people have been arrested in connection with the fatal stabbing of a man in Leicester which police believe may have been linked to the suspected "firebombing" of a nearby house hours later, in which a mother and her three children were killed.
A man and two women, all aged 19, were arrested over the murder of football coach Antoin Akpom, a father-of-one in his 20s.
Shehnila Taufiq died along with her three teenage children hours later at the family's home in Wood Hill about half a mile away in the Spinney Hills area of the city .
Leicestershire Police have begun two murder inquiries, and said they were investigating whether the fire had been intentionally started as a revenge attack.
"The arrests are in connection with an investigation launched on 12 September, after police officers found a man injured in Kent Street," said a statement from Leicestershire police.
"He was taken to Leicester Royal Infirmary where he later died. The man has not yet been formally identified."
The male suspect was arrested in London, police said.
Describing both incidents as "terrible, terrible crimes", assistant chief constable Roger Bannister said: "I can't confirm here and now it is a revenge attack - it may be, it may not be - but lines of inquiry will certainly get to the bottom of that."
Referring to talk of a "firebombing" among neighbours, Bannister said: "Firebombing in my experience means different things to different people. We are working really closely with the fire brigade.
"Exactly how the fire started, where it started and of course who is responsible is something we will get to the bottom of."
Bannister said neither family members nor the male stabbing victim were known to police. The Taufiq family had no involvement in the earlier stabbing, police said.
Officers are working to trace a man who also lived in the property in Wood Hill, whose upstairs windows were still blackened after fire crews recovered the bodies.
The children, named by a local mosque as Zainab, 19, Jamal, 17, and Bilal, 15, died in their bedrooms at their terraced house just after 12.30am on Friday.
Their father Dr Muhammad Taufiq al Sattar, was believed to be on his way home from Ireland, where he works.
Shocked neighbours of the Taufiqs suggested that if the family had been targeted, it might have been a case of mistaken identity.
"Most people feel this was meant for somebody else and they got the wrong house," said a middle-aged neighbour who did not want to be named.
Leicestershire Police added: "The whole area is being carefully examined which may take some time.
"Specially-trained officers will be searching for evidence connected to the incident and this requires careful searching under and around vehicles parked in the street, front gardens and pavements.
"Detectives would like to thank the residents of Wood Hill for their patience and co-operation while this is ongoing."
Extra officers and forensics staff were drafted in to investigate the crimes and bolster patrols in the area.
Leicester-based Imam Ibrahim Mogra, the assistant secretary-general of the Muslim Council of Britain, said: "I did not know them personally but from what people have told me they were a very religious family, very humble, very modest and very generous.
"The mother was a scholar herself and clearly they were a family that was into religion and education and were held in very high esteem by the local community."
Local shopkeeper Sattar Raidhan, who has lived in the area for 35 years, said: "I believe the husband worked as a doctor in Ireland and he came home every weekend to see his family. I can't even imagine how this would happen to a family like that."
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