They have obtained CCTV footage that they believe has captured the chief suspect near Ms Burns’s home during the weekend before her body was found.
And they have also received reports of multiple sightings of the suspected killer around her home in the Ardshanavooly estate in Killarney in the days before her death.
Gardaí fear that Ms Burns was dead for some time before the tragedy was uncovered on Monday afternoon.
They now believe that the pensioner may have been pestered by the chief suspect in the case who, when problems arose because of the chaotic nature of his lifestyle, would often seek help from the kind-hearted grandmother, often in the form of cash handouts.
The man would call to the woman’s home for help.
And he would on occasion also hang around waiting for her if she was not present.
Gardaí are also investigating indications that, on previous occasions when help was refused, threats were issued by the individual.
The revelation came as detectives have made “major progress” with their forensic and technical investigation into the precise circumstances in which the grandmother and mother of four was killed.
The body of Ms Burns was discovered at lunchtime on Monday in the front room of her two-storey terraced home after neighbours had called to the property amid concerns from an overseas relative who had been unable to contact her.
She was lying on her side and had traces of blood on her face.
The woman – hailed as “Killarney’s smiling lady” by neighbours – had been violently assaulted and an attempt had been made to strangle her.
Mrs Burns had marks to both her face and neck.
However, gardaí have refused for operational reasons to release the results of the post-mortem examination that was conducted by assistant State pathologist Dr Margaret Bolster at University Hospital Kerry (UHK) on Tuesday.
Detectives are convinced Ms Burns knew her killer.
There was no sign of forced entry at the property – and there was similarly no sign of a robbery.
Gardaí have conducted exhaustive door-to-door inquires through the entire Ardshanavooly area, an estate on the outskirts of Killarney town centre and only a short walk from the Killarney Outlet Centre.
Officers have also been trawling through hours of CCTV security camera footage recorded from private homes and business premises in the area.
One source revealed that the chief suspect in the case has been placed in the general Ardshanavooly area from multiple sightings – not far from Ms Burns’s home – during the weekend before her body was found.
Detectives also hope to have the results of critical forensic tests over the coming days.
These will include a painstaking analysis of clothing and other items that were recovered from a search of two areas in Killarney town centre on Wednesday, near the Wishing Well and Ross Castle.
All the results will be compiled in a report to be submitted to the Director of Public Prosecutions (DPP).
A man in his 50s who was arrested and questioned for three days by gardaí on suspicion of the murder of the grandmother was released without charge on Thursday.
The man – who was known to the deceased – was arrested in the south Kerry area.
He is described as having “a chaotic lifestyle”.
Ms Burns’s requiem mass will take place at St Mary’s Cathedral in Killarney at 4pm tomorrow. Her funeral cortege will arrive at St Mary’s Cathedral at 2.15pm with a private cremation to follow the requiem mass.
The funeral notice said the 75-year-old was “devastatingly missed and infinitely loved by her children, grandchildren, sisters, brother and the whole of her extended family, friends, neighbours and the wider (Kerry) community”.
Numerous Killarney groups are expected to pay tribute at her funeral to a woman described as kind-hearted and utterly devoted to her family and community.
Gardaí have again urged anyone who may have spotted suspicious activity in the area to contact them and assist their investigation.
“We are appealing to any persons who were in the Ardshanavooly area of Killarney between 5pm on Friday and 1pm on Monday and observed any activity which drew their attention, to come forward,” a spokesperson said.
Ms Burns has four children, several of whom now live abroad.
She also hails from a large and highly respected south Kerry family.
She adored travel and regularly visited European cities, often travelling to meet up with some of her children.