In the k-supplier problem, we are given a set of clients C and set of facilities F located in a metric (C∪F, d), along with a bound k. The goal is to open a subset of k facilities so as to minimize the maximum distance of a client to an open facility, i.e., minS⊆F: |S|=kmax v∈C d(v,S), where d(v,S) = minu∈S d(v,u) is the minimum distance of client v to any facility in S. We present a 1 + √3 < 2.74 approximation algorithm for the k-supplier problem in Euclidean metrics. This improves the previously known 3-approximation algorithm  which also holds for general metrics (where it is known to be tight). It is NP-hard to approximate Euclidean k-supplier to better than a factor of √7 ≈ 2.65, even in dimension two . Our algorithm is based on a relation to the edge cover problem. We also present a nearly linear O(n·log2 n) time algorithm for Euclidean k-supplier in constant dimensions that achieves an approximation ratio of 2.965, where n = |C∪F|.