  The existence of strange matter in compact stars may pose striking sequels of
the various physical phenomena. As an alternative to neutron stars, a new class
of compact stars called strange stars should exist if the strange matter
hypothesis is true. In the present article, we investigate the possible
construction of the strange stars in quark matter phases based on the MIT bag
model. We consider scenarios in which strange stars have no crusts. Then we
apply two types of equations of state to quantify the mass-radius diagram for
static strange star models performing the numerical calculation to the modified
Tolman-Oppenheimer-Volkoff (TOV) equations in the context of $4D$
Einstein-Gauss-Bonnet gravity. It is worth noting that the GB term gives rise
to a non-trivial contribution to the gravitational dynamics in the limit $D \to
4$. However, the claim that the resulting theory is of pure graviton was cast
in doubt by several grounds. Thus, we begin our discussion with showing the
regularized $4D$ EGB theory has an equivalent action as the novel $4D$ EGB in a
spherically symmetric spacetime. We also study the effects of coupling constant
$\alpha$ on the physical properties of the constructed strange stars including
the compactness and criterion of adiabatic stability. Finally, we compare our
results to those obtained from the standard GR.
