Taylor dispersion of a solute in a pulsatile flow of a viscoelastic fluid, whose constitutive equation follows the Maxwell model, through an eccentric annulus is investigated in this work. To determine the effective dispersion coefficient, $\mathscr{D}_{\textit{eff}}$, we have used the multiple-scale analysis in conjunction with the homogenization method. The governing equation describing this dispersive phenomenon for solute concentration is the advection-diffusion equation, which depends on the velocity profile. Therefore, the momentum equation must be solved in advance. A hyperbolic partial differential equation in a bipolar coordinate system was derived by combining the Cauchy momentum equation with Maxwell’s constitutive equation. Parameters such as the Womersley number, ${\textit{Wo}}$, and the Deborah number, ${\textit{De}}$, control the time-dependent flow and viscoelasticity, respectively. For low Womersley numbers, i.e. for low frequencies, an increase in the Deborah number, the eccentricity, $\phi$, and gap width, $\gamma$, leads to an enhancement of the effective dispersion coefficient. For instance, a fluid with ${\textit{De}} = 5$ could increase $\mathscr{D}_{\textit{eff}}$ by two orders of magnitude compared with a Newtonian fluid with the same settings ($\phi = 0.3$ and ${\textit{Wo}} = 0.1$). However, this enhancement due to the viscoelastic effect is only significant at low frequencies. An advection-diffusion equation for the mean concentration in the cross-section was also derived and evaluated in the same low-frequency limit. It was concluded that pulsatile flow maximises the axial dispersion compared with steady and purely oscillatory flows.