Shehnila Taufiq, who was in her 40s, died along with her 19-year-old daughter and sons, aged 17 and 15, when their home was engulfed in flames early yesterday.
Mrs Taufiq's neuro-surgeon husband, Mohammad Sattar, was believed to be en route home from Ireland where he works.
The mother and her children, named by the local mosque in Leicester as Zainab, Jamal and Bilal, died in their bedrooms at their terraced house in Wood Hill in the Spinney Hills area of the city several hours after a man was killed in nearby Kent Street.
Police said they could not rule out a link between the two events and said they are investigating whether the fire had been intentionally started as a revenge attack.
Neighbours reacted with disbelief to the idea they could have been intentionally killed and claimed they may not have been the intended target.
A middle-aged man, who did not want to be named, said: "I didn't know the family personally, but I knew of them and they were a very nice family. Most people feel this was meant for somebody else and they got the wrong house. The family were very well-educated. He was a brain surgeon in Ireland and he brought his family over here to study Islamic literature."
The force launched its first murder inquiry shortly after 5.30pm yesterday when a man in his 20s was taken from Kent Street - half-a- mile from the scene of the fire - to Leicester Royal Infirmary, where he died.
Less than 24 hours later a second inquiry began when neighbours reported flames shooting out of the first floor of the Taufiq family home at 12.30am.
As police appealed for information, Assistant Chief Constable Roger Bannister said officers were investigating a theory that Mrs Taufiq and her children were killed in revenge for the earlier attack.
He said: "These are obviously both very serious incidents and the investigations are in their very early stages to establish if there are any links between them. 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."
He said neither the family nor the man who was killed were known to police.