They belong to a man in his 40s, his wife, in her 30s, and their three sons aged six, 11 and 13.
They have been named locally as Alan and Clodagh Hawe and their sons Niall, Ryan and Liam.
Police say they are not currently looking for anyone else in relation to the deaths.
The alarm was raised at 10:45 BST on Monday, after someone called to the house and could not get in.
They contacted police who searched the house and found two of the bodies downstairs, the other three were found in upstairs bedrooms.
Speaking at a press conference on Monday evening, Assistant Commissioner John O'Driscoll said: "We believe all the answers are in that house.
"We are working along very definite lines of inquiry.
"The most likely scenario is that one person in that house may have caused the deaths of the others."
He said there was no evidence that a firearm had been used and a number of objects had to be examined before a cause of death can be confirmed.
Bishop Leo O'Reilly of Kilmore said the deaths were "shocking" and "near impossible to comprehend".
The bodies of five members of a family have been found at a house near Ballyjamesduff in County Cavan in the Republic of Ireland.