  Future electron-position colliders, such as CEPC and FCC-ee, have the
capability to dramatically improve the experimental precision for W and Z-boson
masses and couplings. This would enable indirect probes of physics beyond the
Standard Model at multi-TeV scales. For this purpose, one must complement the
experimental measurements with equally precise calculations for the theoretical
predictions of these quantities within the Standard Model, including three-loop
electroweak corrections. This article reports on the calculation of a subset of
these corrections, stemming from diagrams with three closed fermion loops to
the following quantities: the prediction of the W-boson mass from the Fermi
constant, the effective weak mixing angle, and partial and total widths of the
Z boson. The numerical size of these corrections is relatively modest, but
non-negligible compared to the precision targets of future colliders. In
passing, an error is identified in previous results for the two-loop
corrections to the Z width, with a small yet non-zero numerical impact.
