In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and ϕ-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connesbiprojective. Next, we show that for a locally compact group G, the Banach algebra M(G) is approximately Connes-biprojective if and only if G is amenable. Finally, for an infinite commutative compact group G, we show that the Banach algebra L2(G) with convolution product is approximately Connes-biprojective, but it is not Connesbiprojective.
