Even though it is not possible to differentiate general relativity from teleparallel gravity using classical experiments, it could be possible to discriminate between them by quantum gravitational effects. These effects have motivated the introduction of nonlocal deformations of general relativity, and similar effects are also expected to occur in teleparallel gravity. Here, we study nonlocal deformations of teleparallel gravity along with its cosmological solutions. We observe that nonlocal teleparallel gravity (like nonlocal general relativity) is consistent with the present cosmological data obtained by SNe Ia + BAO + CC + [Formula: see text] observations. Along this track, future experiments probing nonlocal effects could be used to test whether general relativity or teleparallel gravity gives the most consistent picture of gravitational interaction.