The objective of this study is to examine the possible existence of traversable wormhole geometries within the context of $f (R, L_{m}, T)$ $f (R, L_{m}, T)$ gravity. To meet this objective, we employ the Karmarkar condition to construct the shape function that aids in identifying the wormhole configurations. This developed function is found to satisfy the essential conditions and provides a link between two asymptotically flat spacetime regions. We then assume the Morris–Thorne line element that expresses the wormhole configuration and formulate the anisotropic gravitational equations for a particular minimal matter-spacetime coupled model of the modified theory. Afterward, we develop three solutions and determine their viability by analyzing whether they violate the null energy conditions. Different stability methods are applied to the resulting geometries to explore the acceptance of the considered modified model. We conclude that the developed wormhole structures potentially fulfill the required criteria and thus exist in this modified gravity under all choices of the matter Lagrangian density.