For a Banach algebra B, the set of weakly almost periodic functions on B is denoted by WAP(B*). It is known that amenability of B yields Connes-amenability of WAP(B*)*. The converse is not generally true though.We prove that under certain assumptions, B is amenable if and only if WAP(B*)* is Connes-amenable. As a result, we show that for a reflexive Banach space E with the approximation property, K(E) is amenable if and only if WAP(K(E)*)* is Connes-amenable
