Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$. The $c$-numerical range of an $n\,\times \,n$ matrix $A$ is defined by
$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$ and the $c$-numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying
$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$