Context. The classical Wolf-Rayet (WR) phase is believed to mark the end stage of the evolution of massive stars with initial masses higher than ~25M⊙. Stars in this phase expose their stripped cores with the products of H- or He-burning at their surface. They develop strong, optically thick stellar winds that are important for the mechanical and chemical feedback of massive stars, and that determine whether the most massive stars end their lives as neutron stars or black holes. The winds of WR stars are currently not well understood, and their inclusion in stellar evolution models relies on uncertain empirical mass-loss relations. Aims. We investigate theoretically the mass-loss properties of H-free WR stars of the nitrogen sequence (WN stars). Methods. We connected stellar structure models for He stars with wind models for optically thick winds and assessed the degree to which these two types of models can simultaneously fulfil their respective sonic-point conditions. Results. Fixing the outer wind law and terminal wind velocity ν∞, we obtain unique solutions for the mass-loss rates of optically thick, radiation-driven winds of WR stars in the phase of core He-burning. The resulting mass-loss relations as a function of stellar parameters agree well with previous empirical relations. Furthermore, we encounter stellar mass limits below which no continuous solutions exist. While these mass limits agree with observations of WR stars in the Galaxy, they contradict observations in the LMC. Conclusions. While our results in particular confirm the slope of often-used empirical mass-loss relations, they imply that only part of the observed WN population can be understood in the framework of the standard assumptions of a smooth transonic flow and compact stellar core. This means that alternative approaches such as a clumped and inflated wind structure or deviations from the diffusion limit at the sonic point may have to be invoked. Qualitatively, the existence of mass limits for the formation of WR-type winds may be relevant for the non-detection of low-mass WR stars in binary systems, which are believed to be progenitors of Type Ib/c supernovae. The sonic-point conditions derived in this work may provide a possibility to include optically thick winds in stellar evolution models in a more physically motivated form than in current models.