In this paper, we study different Solar System tests in a modified Teleparallel gravity theory based on an arbitrary function f (T, B) which depends on the scalar torsion T and the boundary term B. To do this, we first find new perturbed spherically symmetric solutions around Schwarzschild for different power-law forms of the arbitrary Lagrangian. Then, for each model we calculated the photon sphere, perihelion shift, deflection of light, Cassini experiment, Shapiro delay and the gravitational redshift. Finally, we confront these computations with different known experiments from these Solar System tests to put different bounds on the mentioned models. We then conclude that f (T, B) is compatible with these Solar System experiments with a wide range of parameters which are relevant for cosmology.