It now seems general fact that when a massive star core collapses at the end of its lifecycle it forms a black hole, can a black hole be formed skipping the star phase altogether?
There are speculated objects known as "primordial black holes" which could have been present at the origin of the universe, or formed in its early stages via presently unknown processes, but there is no observational evidence that such objects exist. Also, some models for forming supermassive black holes involve the coalescence of such a huge amount of matter that you would not call it a star, unless some kind of supermassive superstar.
Alternatively, the mass could form stars first, and then coalesce into the supermassive black hole, but those stars themselves would not necessarily need to collapse first. Indeed, since much of the mass that goes into stars goes into lower mass stars that don't make black holes, it seems likely that a significant fraction, or even a majority, of the mass in supermassive black holes was not in a black hole prior to falling into the supermassive black hole.
Finally, since we don't know much about dark matter, one presumes it might be possible for dark matter to form black holes via some presently unknown mechanism, but there is no evidence dark matter would do that, and the idea that dark matter might itself be already in the form of black holes has not been supported by observational searches.
So I'd say our current expectation is that supermassive black holes are largely made of matter that did not collapse in a massive star, and stellar-mass black holes could be mostly or entirely matter that did.
Here's a possible alternative sequence. This is off the top of my head.
Suppose somehow (and quite unlikely in our universe) there's a large cloud of particles all of whom have atomic number greater than Fe . Over time, this cloud coalesces under the usual laws of gravity. But due to the high atomic numbers, the pressure and density required for fusion cannot be achieved. (In fact, these elements, even if they fuse, do not release energy but rather absorb it). The continued coalescence leads to black-hole collapse.
I repeat: I have not the knowledge of the various conditions necessary for collapse, so this may not be possible. But more or less this is what's described in the link to oup provided in the downvoted answer.
In the very early evolution of the universe, primordial black holes could have formed before there were any stars.
The early universe was very dense and small variations in density could cause regions to undergo gravitational collapse, forming black holes. These black holes would not have formed from a star, and could be outside the range of masses of a black hole formed from the collapse of the core of a massive star. Depending on the model you use for the scale of inhomogeneities in the Early Universe, these black holes could have masses between a milligram up to several thousand times the mass of the sun.
It is possible that some of the supermassive black holes at the centre of galaxies may have been primordial black holes. (but this is speculative)