The original form of the Fuglede-Putnam theorem states that the operator equation AX = XB implies A*X = XB* when A and B are normal. In our previous paper we have relaxed the normality in the hypotheses on A and B as follows: if A and B* are subnormal and if X is an operator such that AX -XB, then A*X -XB*. We shall show asymptotic versions of this generalized Fuglede-Putnam theorem; these results are also extensions of results of Moore and Rogers.