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In this paper, the stability properties of milling processes are presented when the axial component of the cutting force is also taken into account as a compressive force acting on the cutting tool. The mechanism of the material removal operation is described by a one degree-of-freedom oscillator subjected to the so-called surface regeneration effect. This can be represented mathematically by a nonlinear, time-periodic, delay differential equation. The regenerative compression affects the natural frequency of the system, which is modelled through the lateral stiffness variation of the bearing support of the main spindle. This extension of the general milling model leads to the birth of stable islands within the originally unstable parameter domain. Since the exploitation of these regions may open new ways in practice for increasing the efficiency of machining, the domain of attraction is investigated qualitatively near the stability boundaries of the stable island by means of a semi-analytical method.