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In the present article, the authors introduce two new subclasses of holomorphic and bi-univalent functions with respect to the symmetric points defined in the domain of open unit disk $\Delta:=\{z \in\mathbb{C} |z|<1\}$ by making use of subordination between two analytic functions and also using the Gegenbauer polynomials. We investigate bounds of some of the initial Taylor-Maclaurin coefficients belonging to this newly constructed holomorphic and bi-univalent function class. Moreover, we derive the well-known Fekete-Szegö functional for the above said classes. Some of the corollaries of the main results are pointed out.