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It is established that every (not necessarily linear) 2-local$^*$-homomorphism from a von Neumann algebra into a C$^*$-algebra is linear anda $^*$-homomorphism. In the setting of (not necessarily linear) 2-local$^*$-homomorphism from a compact C$^*$-algebra we prove that the sameconclusion remains valid. We also prove that every 2-local Jordan$^*$-homomorphism from a JBW$^*$-algebra into a JB$^*$-algebra is linear and aJordan $^*$-homomorphism.

A Kowalski-S{\l}odkowski theorem for 2-local $^*$-homomorphisms on von Neumann algebras