Many "bouncing cosmologies" (by Lee Smolin, Nikodem Poplawski, and others) substitute the formation of a new "Local Universe" for the singularity mentioned in an earlier answer: Poplawski goes into the most detail, in several papers on the Arxiv website that were written as recently as 2018. These cosmologies generally have the advantage of balancing a contraction (which begins when the star runs out of pressure-producing fuel) against expansion, thereby allowing a universe eternal to the past, which is otherwise impossible (for geometrical reasons described in the Borde-Guth-Vilenkin Theorem).
I have to clarify that the universe eternal to the past would be an ensemble, or set of iterations, of local universes. There is ample astronomical evidence for black holes, including the circular orbits of many lone stars whose binary partner has become one. Most stars are in binary pairs.
I think the OP may be thinking of microscopic black holes, once an idea of Hawking's, for which there is little evidence in our currently-observable region. (The spatial scale of objects outside it is, of course, entirely hypothetical.)
I believe he may have been misinformed on two points: First, that BHs result from external pressure rather than a lack (as I pointed out earlier) of internal pressure, and, second, that the event horizon appears somewhere inside the collapsing object and stays there--it doesn't; it propagates outward from the center of the collapsing star. The ideas of the collapse taking forever are correct only in the inclusion of further BHs within the local universe in formation, more BHs within them, etc., and derive from the "past eternality" permitted by the balancing of such contraction against expansion.
That last statement sounds so outrageous that I feel obligated to back it up with a little math, visibly provided by Vilenkin in his 2013 paper "Arrows of time and the beginning of the universe", a critique of the bouncing cosmology that Carroll and Chen had devised in their own paper "Spontaneous inflation and the origin of the arrow of time": Vilenkin concludes that "an infinite Cauchy surface with random initial conditions will generally produce inflating regions in both time directions", which would, however, "be surrounded by singularities, and...have singularities in their past and future", while the current Wikipedia article "Gravitational singularity" describes singularities as having the "infinite density" and "infinite temperature" I'm eliciting in that statement. (Carroll and Chen don't particularly set their cosmology within any black hole, but Poplawski provides a mechanism which would get a local effect going that could easily--even to inhabitants only average-sized on an infinite range of scales--resemble "our" Big Bang sufficiently to be, for them, as observationally indistinguishable from it as our own appears to be for us. All of this could happen within space appearing to be black, subjectively deep, and laden [by whatever Hawking radiation would've evaporated previous BHs] with enough potential for point-like bosons to get just such a shebang rolling again, in any region that had been rather quiescent for a while.)
The "event horizon" is, since remarks by Hawking in 2014, sometimes referred to as an "apparent horizon", which leaves the door open to whatever evidence might eventually substantiate a beginning for a multiverse. Such evidence might include changes in the CMB data, although our current CMB data, described in the Arxiv paper "Non-parametric reconstruction of an inflaton potential", is consistent with Poplawski's theory.
In case anyone might conclude that I'm implying that a multiverse might consist of a quasi-infinite collection of sub-microscopic, toy-like versions of our observable region, I have to say that that's not necessarily my aim. Although such a representation of it might be consistent with such a scale-invariant theory as GR (especially given Einstein's comments about the speed of light, that can be seen in the 1st paragraph of Section 27 of Wikisource's online version of his 1916 popularization "Relativity: The Special and General Theory"), the effect of time dilation, in the intense gravitational field surrounding the center of any large star undergoing gravitational collapse, might balance any reduction in the spatial scale of material and energetic objects (including both subatomic particles of the star itself and new particles, made real by their separation from partners in virtual pairs, drawn into the collapse by tidal effects) arriving there, while the reversibility of the length contraction in the material ones, during their subsequent outward acceleration from contact with the stellar fermions, might combine with it to leave (or, at least, enhance) that impression of spatial expansion which characterizes our Big Bang.