The phoenix is a bird about which the earliest sources are far more cautious than the later ones.
Herodotus, writing in the fifth century BCE, gives the first extended account. He reports what the Heliopolitans say: a bird resembling an eagle, red and gold, which comes once in five hundred years bearing the body of its parent embalmed in myrrh, to bury it in the temple of the sun. Herodotus adds that he has not seen it himself except in a painting, and that he does not find the story credible. That scepticism is worth noting, because everything afterwards drops it.
The Egyptian source is almost certainly the bennu, a heron associated with Ra, with Atum and with Osiris, connected in the Heliopolitan theology with the primeval mound that first rose from the waters and with the daily rebirth of the sun. The bennu is a real element of Egyptian religion and its associations are with cycle and origin. The Greek elaboration, the five-hundred-year interval, the funerary flight from Arabia, goes well beyond it.
Fire is a later addition. Herodotus has no fire. The self-immolation on a nest of spices, and the young phoenix rising from the ashes of the old, develops in Roman-period writing; Tacitus discusses conflicting accounts of the interval, Pliny reports the bird's appearance in Rome under Claudius, and Ovid and Lactantius give the burning version. By the time the image is stable it has travelled a long way from anything Egyptian.
The number of years is unstable throughout: five hundred in Herodotus, 540, 1,000, 1,461 in others, the last matching the Egyptian Sothic cycle, which is why some accounts tie the bird to the calendar.