A real hypersurface M in a Kähler manifold \(\widetilde{M}\) is called Hopf if the structure vector \(\xi =-JN\) is an eigenvector of the shape operator of M. A Hopf hypersurface in the complex projective space has the property that each parallel hypersurface to a Hopf hypersurface is Hopf. Also any tube over a complex submanifold in the complex projective space is a Hopf hypersurface. Thus Hopf hypersurfaces have a nice geometric property as isoparametric hypersurfaces in real space forms. For real hypersurfaces in Hermitian symmetric spaces of rank greater than 1, the classification of Hopf hypersurfaces such that each parallel hypersurface to a Hopf hypersurface is Hopf, is a fundamental problem. In this paper, we will give a classification of the real hypersurfaces \(M^3\) in \(\mathbb {S}^2\times \mathbb {S}^2\) such that their nearby parallel hypersurfaces are Hopf, i.e., \(M^3\) is locally congruent to either a product hypersurface \(\Gamma \times \mathbb {S}^2\) or a standard example \(M_t\) for some t. Note that there exists a Hopf hypersurface in \(\mathbb {S}^2\times \mathbb {S}^2\) such that each parallel family of the Hopf hypersurfaces along the normal geodesics is not Hopf.
