We study the entrainment effect between superfluid neutrons and charge neutral fluid (called the proton fluid) which is made of protons and electrons in a neutron star interior within the two-fluid formalism and using a relativistic model where baryon-baryon interaction is mediated by the exchange of $\ensuremath{\sigma}$, $\ensuremath{\omega}$, and $\ensuremath{\rho}$ mesons. This model of strong interaction also includes scalar self-interactions. The entrainment matrix and entrainment parameter are calculated using the parameter sets of Glendenning (GL) and another non-linear (NL3) interaction. The inclusion of $\ensuremath{\rho}$ mesons strongly influences the entrainment parameter (${ϵ}_{\text{mom}}$) in a superfluid neutron star. The entrainment parameter is constant at the core and drops rapidly at the surface. It takes values within the physical range.