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.
The children were named by the local mosque as Zainab, Jamal and Bilal.
Emergency crews battled to save the family but the four victims were found dead inside the bedrooms of the terraced house in Wood Hill in the Spinney Hills area of Leicester just after 12.30am today.
Mrs Taufiq's husband, Mohammad, is believed be returning home from Ireland where he works as a heart surgeon, a friend said.
Leicestershire Police launched two separate murder inquiries within 24 hours.
The first began after officers were called at 5.30pm yesterday to reports that a man in his 20s had been assaulted in Kent Street - just half a mile from the blaze which erupted several hours later.
He was taken to Leicester Royal Infirmary where he died.
Today, 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."
Referring to neighbours talking about a "fire-bombing", Mr Bannister added: "Fire-bombing, 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."
Urging witnesses to get in touch, Mr Bannister said it was possible that both offences, which he described as "terrible, terrible crimes", were linked.
He said the crimes might be linked because of the close locations and short space of time between them.
The officer said neither the family nor the man who was killed were known to police.
A mosque attended by the Taufiq family issued a statement expressing shock at the deaths.
A spokesman for the Jame Mosque, about 100 yards from the family's home, said: "Four people residing at Wood Hill have tragically passed away in a house fire.
"This included Shehnila Taufiq, Zainab Taufiq and Jamal and Bilal Taufiq.
"The sons and daughters were aged between 15 and 20. The father was away in Ireland during the week working as a heart surgeon."
The family attended the mosque in Asfordby Street while Mrs Taufiq, Zainab and Bilal were also enrolled on its educational programme.
The spokesman added: "The members of our mosque, the staff and all of our colleagues are extremely shocked and are praying for them.
"We would also like to send our condolences to the father, Dr Taufiq, who, as you can imagine, would be going through a difficult time."
Among friends and neighbours who paid tribute at the scene was teacher Mohammed Qasim Manjra.
Mr Manjra, who taught Jamal history at a nearby independent Islamic school, said: "He was a very sociable kid and we are very sad that he has gone.
"It's a very big shock for the community."
Leicester-based Imam Ibrahim Mogra, the assistant secretary-general of the Muslim Council of Britain, also visited the scene.
He told reporters: "Local people are devastated. It's hard to believe that such a tragedy could occur and four lives could be lost in a matter of minutes.
"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."
Asked for his views on possible links between the fire and the murder committed in the city, Mr Mogra added: "We just have to wait for the police to carry out their investigation and I just pray that there is no link between the two incidents."
Local store-keeper Sattar Raidhan, who has lived in the area for 35 years, said: "The whole family used the shop regularly and I heard this morning that there had been a fire.
"One of the boys - the 15-year-old - came in the shop yesterday. They were a loving family and had a lot of friends.
"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."
Labour MP Keith Vaz, who represents Leicester East, met local people near the police cordon sealing off Wood Hill.
He said: "This is a very peaceful area so this is a big shock to local people and I'm just here to show my support.
"I have spoken to the (Police and Crime) Commissioner, Sir Clive Loader, and to the Chief Constable and I think we need to let the police get on with their job.
"Only when we know the facts can we come to any conclusions as to why this has happened to an ordinary hard-working family.
"The imam has just told me they were very active in the work that they were doing locally.
"The father is a doctor and is being comforted by members of the community... a community who are clearly in grief.
"A whole family wiped out in this way, with only the father remaining, is a big shock and a real tragedy.
"If indeed people have deliberately done this, then I'm sure the police will do everything they can in order to catch them and to bring them to justice.
"We must leave the police to get all the facts together."
The force has drafted in extra officers and forensics staff to probe both crimes and increased patrols in the area.
Several neighbours of the family voiced fears that they had been mistakenly targeted in a revenge attack meant for someone else.
A middle-aged man, who was woken by the sound of neighbours attempting to alert the family, said the heat of the fire was so intense it could be felt on the opposite pavement.
The eyewitness, who asked not be named, also praised the bravery of two men who tried to make their way into the blazing property.
"I didn't know the family personally but I knew of them and they were a very nice family," the man said.
"Most people feel this was meant for somebody else and they got the wrong house."
The neighbour added: "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 lads were very pious and one of them was memorising the Koran and was on the ninth chapter."
Describing the incident itself, the witness said: "I was in bed at about midnight and I heard all the commotion and thought it was drunks or whatever, and then I heard shouting.
"The guy opposite broke the door down but the fire was so intense they couldn't get in.
"They were throwing bricks at the windows at the top of the house to alert them inside but I think it was too late.
"The fire was very intense and you could feel the heat of it across the road.
"The neighbours tried their level best but it was too severe."
Mr Taufiq was a consultant neurosurgeon has been working in Beaumont Hospital in Dublin and with private hospitals in the city, the Blackrock and Hermitage clinics.
"The board and staff of Beaumont Hospital have learned, with shock, of the tragic loss suffered by our colleague Mr Taufiq Sattar," a spokesman said.
"We wish to express our sincere condolences to Mr Sattar and assure him that our support and thoughts are with him at this terrible time."
Colleagues at Blackrock and the Hermitage also expressed their sympathy.
"Colleagues and friends at Blackrock Clinic and Hermitage Clinic have expressed their shock at the fire in Leicester earlier today that tragically took the lives of our dear colleague's wife Shehnila and children Zainab, Bilal and Jamal," they said.
"Our thoughts and prayers are with Mr Sattar at this most difficult time."