White Holes (WH) are a theoretical possibility. It is theoretically possible that a Black Hole (BH) and WH could exist under certain conditions.
First, Einstein-Rosen obtained a general relativity solution where the two are connected by a neck or bridge, called the Einstein-Rosen bridge. Thing is later it was calculated that it could only be stable so that something could go from one side to the other if the bridge is made of so-called exotic matter, matter with negative mass.
Nobody has detected negative mass particles. But there are some theoretical constructs in string theory and other non standard model theories that have the possibility of exotic matter having appeared early after the birth of the universe, and they or some resulting wormhole may be detectable. We have found nothing like this, but we can't yet see (electromagnetically) much closer than 380000 years from the Big Bang, because of the recombination wall. Gravitational. Wave detectors in space, if big enough, could detect gravitational waves from earlier on, and possibly match them to what some of these theoretical constructs imply. So, nothing found yet, but it is not theoretically un-detectable, so semi-fair game.
The same is true with Einstein Cartan theory, with torsion in the gravitational connection (i.e., a non symmetric connection), where a wormhole may be possible. Nothing found yet supports Einstein-Cartan theory, but not fully disproven either.
So, yes, they are possible, though no evidence and maybe it just does not happen. Papers and so on still come out on it once in a while. Some people are studying it where it could be the beginning g of the universe with some models they propose about string theory M-branes pre Big Bang that led to it.
Not speculative, but far from any real good reason to think they are there somewhere. See the wiki article at https://en.m.wikipedia.org/wiki/White_hole