  In this paper, using a Bestest Little Higgs Model (BLHM) approach, we obtain
bounds on the top-quark electromagnetic and weak dipole moments. These physical
observables are sensitive to quantum corrections induced by virtual particles
at the one-loop level. The contributions of the virtual particles are obtained
from the vertices of scalar bosons, vector bosons, and heavy quarks: $tq_iS_i$,
$S_i=h_0, H_0, A_0, \phi^{0}, \eta^{0}, \sigma, H^{\pm}, \phi^{\pm},
\eta^{\pm}$; $tq_iV_i$, $V_i= \gamma, Z, W, Z', W'$; $V q_i\bar q_i$,
$V=\gamma,Z$, $q_i=b, t, B, T, T_5, T_6, T^{2/3}, T^{5/3}$; $ V
W^{-}_{i}\Phi^{+}_i $, $W_{i} \equiv W, W'$, $\Phi^{\pm}_{i}= \phi^{\pm},
\eta^{\pm}$; $ZZH_i$, $H_i=h_0, H_0$; $ZZ' H_i$; and $VW'W'$. With these new
contributions, we evaluated the electroweak dipole moments $ a_t$, $ a^{W}_t $,
$ d_t $ and $ d^{W}_{t} $ of the top-quark in the diagonalization schemes $y_2
> y_3$ and $y_2 < y_3$ to the Yukawa couplings. For the BLHM parameters, we
choose the following values of $m_{A_0}= 1000$ GeV, $m_{\eta^0}= 100$ GeV,
$f=[1000, 3000]$ GeV, $F=[3000, 6000]$ GeV and $\tan\beta=3$, which give the
best sensitivity on the electroweak dipole moments of the top-quark in this
context: $a_t=1.39 \times 10^{-4} + i\, 6.55 \times 10^{-5}$; $a^{W}_t=6.31
\times 10^{-5} - i\, 1.39 \times 10^{-5}$ for the $y_2 > y_3 $ scenario, while
$a_t= 2.12 \times 10^{-4} + i\, 4.49 \times 10^{-5}$; $a^{W}_t= 2.13 \times
10^{-4} - i\, 1.34 \times 10^{-5}$ for the $y_2 < y_3 $ scenario.
