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The properties of neutron stars are investigated by including δ meson field in the modified quark-meson coupling model. It is found that δ meson has opposite effects on hadronic matter with or without hyperons: it softens the EOSs of hadronic matter with hyperons, while it stiffens the EOSs of pure nucleonic matter. Moreover, by replacing the isovector mesons with the contact scalar-isovector interaction, a considerable change of proton fraction is observed. It is shown that the contact scalar-isovector interaction provides an approach to satisfy the constraint of Direct Urca critical star masses. However, the inclusion of δ meson increases the proton fraction, consequently it decreases the MDU which departs farther from the Direct Urca constraint. Furthermore, the inclusion of δ meson field can increase the maximum mass of neutron star and enlarge the corresponding radius, while these quantities are decreased when the contact scalar-isovector interaction is included. [ABSTRACT FROM AUTHOR]