[en] Enhanced fluctuations from electromagnetic heat flux instabilities in collisionless plasmas may, through wave-particle scattering, constrain the electron heat flux q_e which flows parallel to the background magnetic field. This manuscript examines the linear Vlasov theory predictions for threshold conditions on the whistler heat flux instability at high β_e, the electron beta, a situation which may have application to both solar wind and astrophysical plasmas. Theory predicts a threshold heat flux which scales approximately as β^-0.9_e. One possible consequence of this instability is that it imposes an upper bound on q_e. We discuss possible applications of such a heat flux limit in the cooling flows of clusters of galaxies with β_e>>1. (c) (c) 2000. The American Astronomical Society