The blaze broke out at about 9pm and was already “raging” when six fire crews were dispatched. The cause of the fire was not known.
The burnt-out property, across from Abbotsinch Retail Park, remained under police guard on Saturday as fire crews also remained at the scene.
Neighbours tried to help while the mum and children were trapped inside.
Firefighters arrived within minutes of the first call and attempted to rescue the mum and her children, residents said.
One man who raced to the house to help said Fiona, a pupil at Mary Russell School and the boys, pupils of St Catherine’s Primary, were happily playing outside on the bouncy castle in the days leading up the tragedy.
He said: “It was 8.50pm, that’s when I smelt the smoke. However my neighbour around the corner was having a bonfire and a drink in the garden so I thought it was coming from there.
“By the time I realised what was happening and got round there, the fire brigade was already there.
“I don’t know how to describe it, it was a raging fire, it was moving so quickly.”
The man said he helped distressed neighbours move their cars so more fire engines could access the scene.
He added: “I could hear the firefighters shouting. One of the kids came out first and then an adult female.
“It was absolutely horrible. It’s such a tragedy. I saw the kids out playing just a few days ago.”
On her Facebook page, Ms Daley describes herself as the “proud mum of Fiona, Alexander James and Philip who I love very much”.
A Police Scotland spokesperson said: “Police Scotland officers are continuing enquiries following a fatal fire at a flat in Paisley on Friday, 19 June, 2020.
“Around 9 pm on Friday, 19 June, 2020 emergency services attended a report of fire at an upper cottage flat in Renfrew Road, Paisley.
“Three children, Fiona Gibson, 12, Alexander James Gibson, eight, and Philip Gibson, five, were taken by ambulance to the Queen Elizabeth University Hospital but sadly died a short time after admission.
“A report will be sent to the Procurator Fiscal.
“A 39-year-old woman was taken by ambulance to the Royal Alexandra Hospital in Paisley. Hospital staff describe her condition as critical.”
The spokesperson added: “An investigation is underway to establish the exact circumstances of this incident.
“Anyone who has information that could assist our enquiries should contact police via telephone number 101 quoting incident number 3669 of 19 June 2020.”
A spokeswoman for the Scottish Fire and Rescue Service added: “We were called at 9pm on June 19 to reports of a dwelling on fire in Paisley.
“Operation control mobilised six appliances to the town’s Renfrew Road where firefighters were met by a well-developed fire within a first floor property.
“Fire fighters remain at the scene.”
The announcement that three children had died and a woman was fighting for her life left residents shocked.
Tributes were paid to the victims on Facebook.
One person wrote: “Canny not believe this was just looking at the picture juile put up with wee Philip lying on her after a good day in the sun with his cousins rip precious baby’s.
Another person wrote: “So sad we drove past today and its deeply saddened to find this out that 3 angels have been taken too soon fly high kids hope mum is okay.”
And a third suggested Ms Daley hasn’t been told her children are dead, writing: “That poor woman has got to come through this and find out about the kids it’s so heartbreaking.”