Given a Hausdorff compact space $X$, we study the $C^{*}$-(semi)-norms on the algebraic tensor product $A \otimes _{alg, C(X)} B$ of two $C(X)$-algebras $A$ and $B$ over $C(X)$. In particular, if one of the two $C(X)$-algebras defines a continuous field of $C^{*}$-algebras over $X$, there exist minimal and maximal $C^{*}$-norms on $A \otimes_{alg,C(X)}B$, but there does not exist any $C^{*}$-norm on $A \otimes _{alg,C(X)}B$ in general.