Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $λ_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$ denotes the distance matrix of $G$. In this paper, we verify a upper bound for $λ_1(D(G))$ in a connected graph $G$ to guarantee the existence of a spanning $k$-tree in $G$.