The thermal response of geothermal heat exchangers after 25, 50, or even 100 years of operation plays a fundamental role in their correct sizing. A possible approach is to use time-periodic behavior to approximate long-term behavior, the reason why the present work analyzes the harmonic thermal response of a vertical geothermal borehole to subannual harmonic excitations. The use of asymptotic expansion techniques reveals the existence of a rich asymptotic structure composed of a quasi-steady inner region and an unsteady outer region, both with negligible axial heat conduction, and two asymptotic boundary layers located close to the borehole endings in which axial heat conduction is important. The resulting asymptotic expansion of the problem leads to theoretical models that correctly reproduce the temperature distributions along the borehole, and whose accuracies are comparable to those of the best performing theoretical/numerical models available nowadays.