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OpenBSD i386 supports "all CPUs compatible with the Intel 80486 or better, with Intel-compatible hardware floating point"

NetBSD i386 says "Any i486 or better CPU should work - genuine Intel or a compatible such as Cyrix, AMD, or NexGen."

If you are wondering "why drop support for the 386?," the diff for the removal of 386 support from the Linux kernel is a good read:

tl;dr atomic instructions XADD and CMPXCHG, and serious bugs in many 386 CPUs. Also BSWAP was added in the 486 and speeds some things up a lot.