Abstract <p>The Dirichlet problem for a linear non-homogeneous second-order partial differential equation of elliptic type with a small parameter at the highest derivatives and a singular line (circle) is studied. A sufficient and necessary condition is found under which an intermediate boundary layer arises in the neighborhood of the singular circle in a singularly perturbed problem described by second-order partial differential equations. Using a modified boundary function method, a complete asymptotic expansion of the solution is constructed in the form of an asymptotic series in the sense of Erdelyi. The resulting expansion is justified, i.e., an estimate is obtained for the remainder term.</p>