Some black holes may be so ancient that they predate the stars themselves, forming instead in the chaotic first moments after the Big Bang. There might even be some black holes out there from the universe before the Big Bang.
This is one of those stories with a ton of "ifs" and some fairly daring leaps of logic, but nothing about it is actually impossible. So, with those caveats in mind, let's look at the actual theory. Let's assume that the lifespan of the universe is cyclical, beginning with a Big Bang and a period of frenzied expansion, which eventually leads to a similar phase of contraction into what is ultimately the Big Crunch, which then leads to another Big Bang.
Based on what we know about dark energy and the accelerating expansion of our universe, it doesn't seem terribly likely that we're headed for a Big Crunch, but let's assume for the sake of argument that our Big Bang really did follow on from the Big Crunch of a previous universe. A key thing to keep in mind is that a Big Bang and a Big Crunch would look and act almost exactly the same, except of course they happen in opposite directions.
One of the more intriguing outgrowths of thinking about the physics of the Big Bang is the notion of primordial black holes. Unlike regular black holes, which form out of the remnants of collapsed stars, primordial black holes would have formed in the immediate aftermath of the black hole, when the universe was very small and thus very dense. Parts of this ancient maelstrom would have been particularly dense, and these pockets would have formed into small black holes, which then would have spread throughout the universe.
It's a neat idea, but primordial black holes remain strictly theoretical. But that hasn't stopped physicists Bernard Carr and Alan Coley from taking these two ideas and knitting them together into one spectacular proposal: that there are primordial black holes that formed not in the Big Bang at the beginning of our universe but in the Big Crunch at the end of the one before.
Carr and Coley calculate that black holes of surprisingly reasonable mass could have survived the Big Crunch/Bang - black holes with as much mass as our Sun could have survived the journey from one universe to the next. This means that there really could be objects in our universe that are older than the universe.
But there's a problem, at least if you're interested in some actual hard evidence for all this. We haven't seen a primordial black hole, but astrophysicists have a decent idea of what they should look like - in fact, primordial black holes are one of the dozens of potential candidates to explain the extremely powerful gamma ray bursts that are occasionally seen. The thing is, these pre-universe black holes would look almost exactly the same as regular primordial black holes, and we currently have no way to detect them...again, assuming they exist in the first place.
So for now, black holes from before the Big Bang will probably need to remain just a theory. But hey - it's still one hell of a theory.