Following A. Gray [Tohoku Math. J., II. Ser. 28, 601–612 (1976; Zbl 0351.53040)], an almost Hermitian manifold \((M,J,g)\) is said to be an \(R_2\)-manifold if its Riemann curvature tensor \(R\) satisfies the condition: \(R(X,Y,Z,T) = R(JX,JY,Z,T) + R(JX,Y,JZ,T) + R(JX,Y,Z,JT)\) for any vector fields \(X,Y,Z,T\) on \(M\). The authors suggests to consider the class of \(cR_2\)-manifolds as those of which the Weyl conformal curvature tensor \(W\) satisfies the condition of above. He finds necessary and sufficient conditions for a curvature type tensor to realize the above condition. He discusses certain examples of \(cR_2\)-manifolds which are not \(R_2\)-manifolds.