In the current paper, we introduce the concepts of left ϕ-approximate Connes-amenability and left character approximate Connes-amenability of a dual Banach algebra A that ϕ is a ω *-continuous homomorphism from A to C. We also characterize left ϕ-approximate Connes-amenability of A in terms of certain derivations and study some hereditary properties for such Banach algebras. Some examples show that these new notions are different from approximate Connes-amenability and left character Connes-amenability for dual Banach algebras,