This probably should be a comment, because it addresses the nature of your question rather than makes an attempt to answer it, but I didn't want to splice the following through several connected comments, where the formatting would be less easy to follow.
I think you need to be a lot more precise as to what you mean by "multiverse", as there have been speculations ad nauseam for centuries (and probably for millennia) about our universe being one reality among many. Such speculations became especially prevalent, and a little more founded on mathematical/physical principles, beginning in the late 1800s when fourth dimensional space possibilities became generally known to the educated public (Abbott's 1884 book Flatland, Charles Hinton's extensive writings in the 1880s, etc.). Then there was Einstein's relativity (1900s and 1910s, and extensively popularized in the 1920s), the rise of pulp science fiction magazines (1920s onward), etc.
Below are excerpts from a couple of things I know about off the top of my head (the H. P. Manning book and the Hasse SF story are among the top 10-15 memorable items I read during my early teen years) and a couple of things I found almost immediately by googling (e.g. the Newcomb articles), probably none of which fit what you're looking for, but I strongly suspect Bohm would have been aware of these kinds of speculations. Thus, you should probably edit your question in a way that excludes these types of speculations.
Modern mathematical thought by Simon Newcomb (1894)
(from p. 105) Add a fourth dimension to space, and there is room for an indefinite number of universes, all alongside of each other, as there is for an indefinite number of sheets of paper when we pile them upon each other.
The philosophy of hyper-space by Simon Newcomb (1898)
(from p. 4, right column) Our conclusion is that space of four dimensions, with its resulting possibility of an infinite number of universes alongside of our own, is a perfectly legitimate mathematical hypothesis.
(from p. 136) $[\ldots ]$ similarly, a universe of four dimensions would of necessity contain an infinite number of universes such as ours.
(from p. 44, right column) HOW many of the infinitely smaller atomic cycles I have passed into, I do not know. I tried to keep count of them at first, but somewhere between twenty and thirty I gave it up; and that was long ago. Each time I would think: “This cannot go on forever—it cannot; surely this next time I must reach the end.” But I have not reached the end.
No, he didn't.
In fact he is known for coming up with his own interpretation of QM that relies on a non-local pilot wave. It's usually referred to as de Broglie-Bohmian mechanics as de Broglie, earlier, had similar ideas.
He is also known for developing a notion of implicate and explicate order in QM, writing:
I propose that each moment in time is a projection of the total implicate order.
This new insight can be best described as Undivided Wholeness in Flowing Movement. This view implies that Flow is in some sense prior to that of things which seem to dissolve in this flow.
This is basically a kind of ontological holism and emphasises becoming over being (though perhaps not of Being itself, given Bohms interest in Advaita Vedantism).
Basil Hiley, a coworker of Bohm said:
Bohm says in his quantum theory book, the original one, that quantum mechanics is a mis-nomer. It should be called quantum non-mechanics .... it's nothing like that. It's not mechanism. It's organicism. Nature is more organic than we think it is. And then you can understand why life arose, because if nature is organic, it has the possibility of life in it.
This itself is an old idea. According to Bertrand Russell, this is one reason why Thales conceived the world as water, as water has the possibility of life - it is seminal. Moreover, the idea of nature, that is the universe, is organic and has the nature of an organism was explicitly mentioned in Aristotle.
All this shows that Bohm was far from the notion of a multiverse - he was thinking in terms of an entirely different category to what is common today - at least in physics - but perhaps, not elsewhere.