(Warning: everything that follows is 100% apocryphal and non-canon)
The enigmatic Sirens of Time are the mutated offshots of ancient beings known as the Harpies of Rassilon, which were used by this almighty personage for purposes similar to the Reapers'... but where do the Harpies of Rassilon come from? Why do they always appear as as human-looking females? Why do they often seem to be a single gestalt-like entity? And why do they appear to have some mysterious, existential link to Time Lords?
The Harpies of Rassilon came into existence during the early times of Time Lord society, during the reign of Rassilon. Rassilon had a daughter named Alecta, who was completely devoted to him and who offered herself as the needed subject for one of her father's great experiments. He wanted to see if a Time Lord's twelve incarnations could be dissociated from one another and if the resultant being could live in twelve different parallel timelines while still retaining a form of individual awareness, i.e. a "higher single self" uniting the twelve dissociated personas into a single, gestalt-like entity (with, of course, the development of some trans-temporal powers as a result of this dissociation process). The experiment went terribly wrong - or perhaps worked "too well", resulting in the Harpies of Rassilon, twelve transtemporal "sister-selves" of the same entity - but with no trace left of Alecta's memories, identity and sense of self. And as "impossible beings" (or, if you will, "paradoxical" ones), they did have some worrying time-warping powers, as well as a ravening HUNGER for temporal energies - especially the ones resulting from paradoxes, chronological divergences and similar alterations.
At first, Rassilon managed to control "them" (or "her"?), using the Harpies as temporal "trackers" and "punishers", not unlike his personal pack of time-hounds. Then they faded into legend and oblivion, like many other secrets from Rassilon's first reign. Perhaps their creator / master / father had eventually imprisoned them into some form of time bubble or extra-temporal zone... and they were forgotten. But now that Gallifrey is no more (in my campaign, it genuinely HAS been destroyed during the Time War), they have found a way back in the time streams, haunting the great temporal ocean as the Sirens of Time...