  We study the capabilities of future electron-positron Linear Colliders, with
centre-of-mass energy at the TeV scale, in accessing the parameter space of a
$Z'$ boson within the minimal $B-L$ model. We carry out a detailed comparison
between the discovery regions mapped over a two-dimensional configuration space
($Z'$ mass and coupling) at the Large Hadron Collider and possible future
Linear Colliders for the case of di-muon production. As known in the literature
for other $Z'$ models, we confirm that leptonic machines, as compared to the
CERN hadronic accelerator, display an additional potential in discovering a
$Z'$ boson as well as in allowing one to study its properties at a level of
precision well beyond that of any of the existing colliders.

High Energy Physics - Phenomenology