The degree distance $(DD)$, which is a weight version of the Wiener index, defined for a connected graph $G$ as vertex-degree-weighted sum of the distances, that is, $DD(G)=\sum_{\{u,v\}\subseteq V(G)}[d_G(u)+d_G(v)]d(u,v|G)$, where $d_G(u)$ denotes the degree of a vertex $u$ in $G$ and $d(u,v|G)$ denotes the distance between two vertices $u$ and $v$ in $G$. In this paper, we establish two upper bounds for the degree distances of four sums of two graphs in terms of other indices of two individual graphs.