Suppose n colored points with k colors in are given. The Smallest Color-Spanning Ball (SCSB) is the smallest ball containing at least one point of each color. As the computation of the SCSB in metric () is time-consuming, we focus on approximately computing the SCSB in near-linear time. Initially, we propose a 3-approximation algorithm running in time. This algorithm is then utilized to present a -approximation algorithm with the running time of . We improve the running time to using randomized techniques. Afterward, spanning colors with two balls is studied. For a special case where , there is an algorithm with time. We demonstrate that for any under the assumption that SETH is true, no approximation algorithm running in time exists for the problem even in one-dimensional space. Nevertheless, we consider the metric where a ball is an axis-parallel hypercube and present a -approximation algorithm running in time which is remarkable when k is large. This time can be reduced to when .