A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional $|a_3 - \mu a_2^2|$ for certain classes of univalent functions. In this paper, we consider the class $\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions which is defined by using subordination for analytic functions $\varphi$ and $\psi$ on the unit disc. Our main object is to provide bounds of the quantity $a_3 - \mu a_2^2$ for functions $f(z) = z + a_2 z^2 + a_3 z^3 + \dotsb$ in $\mathcal{C}(\varphi, \psi)$ in terms of $\varphi$ and $\psi$, where $\mu$ is a real constant. We also show that the class $\mathcal{C}(\varphi, \psi)$ is closed under the convolution operation by convex functions, or starlike functions of order $1/2$ when $\varphi$ and $\psi$ satisfy some mild conditions.