  This paper aims to provide new insights on the origin of the TeV source VER
J1907+062 through new high-quality radio observations. We used the Karl G.
Jansky Very Large Array (VLA) to observe the whole extension of VER J1907+062
at 1.5 GHz with a mosaicking technique and the PSR J1907+0602 in a single
pointing at 6 GHz. These data were used together with $^{12}$CO and atomic
hydrogen observations obtained from public surveys to investigate the
interstellar medium in the direction of VER J1907+062. The new radio
observations do not show any evidence of a pulsar wind nebula (PWN) driven by
the pulsars present in the field and no radio counterpart to the proposed X-ray
PWN powered by PSR J1907+0602 is seen in the new VLA image at 6 GHz down to a
noise level of 10 $\mu$Jy/beam. Molecular clouds were discovered over the
eastern, southern, and western borders of the radio shell of G40.5$-$0.5,
suggesting an association with this supernova remnant. We explored several
scenarios for the origin of VER J1907+062. We propose as the most probable
scenario one in which the TeV emission is produced by two separated gamma-ray
sources located at different distances: one of leptonic origin and associated
with a PWN powered by PSR J1907+0602 at $\sim 3.2$ kpc and another of hadronic
origin and produced by the interaction between G40.5$-$0.5 and the surrounding
molecular gas at $\sim 8.7$ kpc.
