The bodies of Seema Banu (37), six-year-old Faizan and his sister Asfira (11) were found at a house in Ballinteer in south Dublin on Wednesday.
They are believed to have been strangled.
The suspect (36) has already had two "voluntary interactions" with gardaí in the presence of his solicitor, with senior sources saying he has been "less than cooperative".
Gardaí have not yet obtained enough evidence to make an arrest.
The man has twice met detectives by arrangement.
The first meeting was on Wednesday night and they a also met yesterday afternoon, but nothing of "major evidential value" has been obtained.
"The investigation team is being very thorough about how it is going about its business and the post-mortems have not yet been completed," a senior source told the Herald.
"There are many strands to this investigation, and apart from definitively establishing the causes of deaths, there is a massive forensic examination still ongoing at the property in Ballinteer, which has so far failed to yield enough evidence to make an arrest.
"There is also an issue in relation to good quality CCTV in the location and the fact that the suspect in the case has made no admissions.
"All this means at the moment is that he is a free man, but that does not mean he is not under constant watch until gardaí are in a position to proceed.
"There is a fear he could try and leave the country."
It can also be revealed that Forensic Science Ireland (FSI) are involved in the case.
The team of specialists are examining three ligatures found at the scene at Llewellyn Court, Ballinteer, shortly before midday on Wednesday which are suspected of having been used to strangle the victims.
Gardaí last night renewed their appeal for information in the case as officers continued to conduct door-to-door enquiries in the estate.
"Post mortems by Dr Heidi Okkers, locum assistant state pathologist, on the three deceased remain ongoing and will be completed tomorrow," a spokesman said.
Gardaí continued to appeal for members of the public not to circulate comments about the case on social media platforms, describing such commentary as uninformed and unhelpful to the investigation.
It has been learned the man's meeting with gardaí yesterday afternoon was made by appointment with a solicitor.
It came after a garda alert was issued over his whereabouts. It described him as being possibly armed and warned that he should not be approached.
Concerns have been raised about the welfare of the man, who had been staying in a city centre hotel after the bodies were found.
"Gardaí had been in contact with him through a legal representative and a meeting was facilitated yesterday," the spokesman said.
"The meeting took place at a location near the city centre."
Two detectives spent more than an hour taking an account from the man before he was told he free to go as he was not formally under arrest.
While he was not detained, the statement he provided can be used at a later stage in the investigation if necessary.
Neighbours become concerned for the family after they had not been seen for a number of days. Members of the Armed Support Unit (ASU) were called in to enter the property, where they discovered Ms Banu and her two children.
Ms Banu's body was found lying face up on a bed in an upstairs bedroom.
The bodies of Faizan and Asfira, who attended the nearby Ballinteer Educate Together National School, were found in a separate bedroom.
It is understood that when gardaí arrived at the scene, the house was overflowing with water after a tap had been turned on in an upstairs bathroom.
Local people held a candlelit vigil in memory of Ms Banu and her children on Wednesday night.