All four at the property on Staines Road, Bedfont, were thought to be from the same family, said the force, according to BBC.
Police said they were called at 15:12 BST on Friday by a person "concerned about the welfare of the occupants".
After officers carried out a "forced entry" they discovered all four bodies.
The force described it as "a terrible incident" and revealed that no other parties were currently being sought.
Speaking at the scene, Ch Supt Sean Wilson, policing commander in west London, said "urgent inquiries" were under way to locate and inform their next of kin.
"I believe we know who they are - four members of the same family - however, urgent inquiries are ongoing to locate and inform their next of kin and it would be inappropriate for me to name them at this time," he told reporters:
Post-mortem examinations are being arranged to establish their cause of death, he added.
It is understood that the property where the bodies were found comprises of two flats and they were discovered in the upstairs property.
A man who lives next door, who did not wish to be named, said: "They were just normal neighbors - we just had greetings here and there, that kind of stuff."
A female neighbour - who also did not wish to be named - said she exchanged pleasantries with the family and revealed the woman was a "lovely mother with two young children".
She told the BBC the family were "absolutely gorgeous".
SKH/PR