Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 to have pointwise 1-type Gauss map.