This paper introduces a general variable neighborhood search (VNS) framework specifically designed for the tree t-spanner problem ( \(T\_t\) -SP). The objective is to determine a spanning tree in a connected, undirected, edge-weighted graph while minimizing the stretch factor (t). The \(T\_t\) -SP is an \(\mathcal{N}\mathcal{P}\) -hard problem and finds significant applications in network design. The proposed VNS incorporates three problem-specific neighborhood operators defining the neighborhood structures. A local search strategy based on domain knowledge is applied to each solution obtained by the corresponding neighborhood. In addition, a re-start criterion is applied to increase the chances of discovering new high-quality solutions. The experimental findings indicate that the proposed VNS delivers performance comparable to the two leading existing methods across the tested instances. Results also reveal that VNS is able to find 13 new values out of 48 instances, establishing it as an effective and promising approach for the \(T\_t\) -SP.