In this paper, we study spherically symmetric traversable wormholes (WHs) in the torsion and matter coupling gravity formalism. We have thoroughly investigated the field equations for WH solutions and found the violation of null energy conditions in the throat vicinity. More specifically, we present an intriguing new class of solutions for static and spherically symmetric WHs that obey the necessary metric conditions by considering specific choices for some significant WHs models constructed from different assumptions for their matter content. They exhibit similar behavior to that found in earlier work on WHs, which is also the case with our solutions for the energy density of these objects. In summary, exact WH models can be traversable if they are filled with exotic matter which violates the energy conditions throughout spacetime while exhibiting a positive definite energy density.