  The top quark plays a central role in many New Physics scenarios and in
understanding the details of Electro-Weak Symmetry Breaking. In the short- and
mid-term future, top-quark studies will mainly be driven by the experiments at
the Large Hadron Collider. Exploration of top quarks will, however, be an
integral part of particle physics studies at any future facility and an $e^+
e^-$ collider will have a very comprehensive top-quark physics program. We
discuss the possibilities of testing NP in the top-quark sector within a
composite Higgs scenario through deviations from the Standard Model in top pair
production for different Centre-of-Mass energy options of a future $e^+e^-$
machine. In particular, we focus on precision studies of the top-quark sector
at a CM energy ranging from 370 GeV up to 3 TeV.
