In this paper, we introduce the concept of transposed BiHom-Poisson (abbr. TBP) algebras which can be constructed by BiHom-Novikov-Poisson algebras. Several useful identities for TBP algebras are provided. We also prove that the tensor products of two (T)BP algebras are closed. The notions of BP 3-Lie algebras and TBP 3-Lie algebras are presented and TBP algebras can induce TBP 3-Lie algebras by two approaches. Finally, we give some examples for the TBP algebras of dimension 2.