  We obtain bounds on the anomalous magnetic and electric dipole moments of the
$t$-quark from a future high-energy and high-luminosity linear electron
positron collider, such as the CLIC, with unpolarized and polarized electron
beams which are a powerful tool to determine new physics. We consider the
processes $\gamma e^- \to \bar t b\nu_e$ ($\gamma$ is the Compton
backscattering photon) and $e^+e^- \to e^-\gamma^* e^+ \to \bar t b\nu_e e^+$
($\gamma^*$ is the Weizsacker-Williams photon) which are one of the most
important sources of single top quark production. For the systematic
uncertainties of $\delta_{sys}=0\%,\hspace{1mm}5\%$, $b-\mbox{tagging
efficiency}=0.8$, center-of-mass energy of $\sqrt{s}=3\hspace{0.8mm}TeV$,
integrated luminosity of ${\cal L}=2\hspace{0.5mm}ab^{-1}$ and
$2\sigma\hspace{1mm}(3\sigma)$ C.L. the future $e^+e^-$ collider may put bounds
on the electromagnetic dipole moments $\hat a_V$ and $\hat a_A$ of the top
quark of the order of ${\cal O}(10^{-2}-10^{-1})$, which are highly competitive
with those recently reported in previous studies.
