In the present paper, we give upper and lower bounds for the spectral norm of $g$-circulant matrix, whose the first row entries are the classical Horadam numbers $U_i^{(a,b)}$. In addition, we also establish an explicit formula of the spectral norm for $g$-circulant matrix with the first row $([U_0^{(a,b)}]^2,[U_1^{(a,b)}]^2, \cdot, [U_{n-1}^{(a,b)}]^2)$.