The authors study the singularly perturbed Volterra integral equation \[ \epsilon u(t)=\int^{t}_{0}K(t-s)F(u(s),s) ds,\quad t\geq 0, \] where \(\epsilon\) is a small parameter, with the objective of developing a methodology that yields the appropriate ''inner'' and ''outer'' integral equations, each of which is defined on the whole domain of interest. The ansatz for the asymptotic representation of the solution is \(u(t)=\sum^{\infty}_{j=0}\epsilon^{m_ j}(y_ j(t)+z_ j(\tau))\), where the \(y_ j\) give the outer solution and the \(z_ j\) comprise the rapidly varying inner solution and \(\tau t/\epsilon^{\alpha}\). It is assumed that \(K(t)=k(t)/t^{\beta}\) where \(0\leq \beta <1\), k(0)\(\neq 0\) and k is sufficiently differentiable. Several examples are presented, but no rigorous proof of the validity of the scheme used is given.