Fire services were alerted to the emergency at the Falcon's View complex near Blanchardstown Shopping Centre around 4.30pm on Sunday.
A source said: "It appears to have begun on a balcony before spreading.
"With these things, the first thing is the preservation of life and putting out the fire.
"So the investigation is really only getting underway, but it shouldn't take too long to establish the cause."
Local TD and government minister Jack Chambers told Morning Ireland on RTE that it was understood that the "fire began on a balcony and spread externally", which would be assessed by the fire service and gardai as part of separate investigations.
The apartment building and a nearby hotel were evacuated and several families could not return home overnight.
As many as 35 apartments are feared to have been affected by the fire.
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