For cross sectionmeasurements, an accurate knowledge of the integrated luminosity is required. The FCC-ee physics programme at and around the Z pole sets the ambitious precision goal of 10(-4) on the absolute luminosity measurement and one order of magnitude better on the relative measurement between energy scan points. The luminosity is determined from the rate of Bhabha scattering, e(+)e(-) -> e(+) e(-), where the final state electrons and positrons are detected in dedicated monitors covering small angles from the outgoing beam directions. The constraints on the luminosity monitors are multiple: (i) they are placed inside the main detector volume only about 1m from the interaction point; (ii) they are centred around the outgoing beam directions and do not satisfy the normal axial detector symmetry; (iii) their coverage is limited by the beam pipe, on the one hand, and by the requirement to stay clear of the main detector acceptance, on the other; (iv) the steep angular dependence of the Bhabha scattering process imposes a precision on the acceptance limits at about 1 mu rad, corresponding to an absolute geometrical precision of O(1 mu m) on the monitor radial dimensions; and v) the very high bunch-crossing rate of 50MHz during the Z-pole operation calls for fast readout electronics. Inspired by second-generation LEP luminosity monitors, which achieved an experimental precision of 3.4 x 10(-4) on the absolute luminosity measurement (Abbiendi et al. in Eur Phys J C 14:373-425, 2000), a proposed ultra-compact solution is based on a sandwich of tungsten-silicon layers. A vigorous R&D programme is needed in order to ensure that such a solution satisfies the more challenging FCC-ee requirements.