We present an extensive study of the ferromagnetic heavy fermion compound U$_4$Ru$_7$Ge$_6$. Measurements of electrical resistivity, specific heat and magnetic properties show that U$_4$Ru$_7$Ge$_6$ orders ferromagnetically at ambient pressure with a Curie temperature $T_{C} = 6.8 pm 0.3$ K. The low temperature magnetic behavior of this soft ferromagnet is dominated by the excitation of gapless spin-wave modes. Our results on the transport properties of U$_4$Ru$_7$Ge$_6$ under pressures up to $2.49$ GPa suggest that U$_4$Ru$_7$Ge$_6$ has a putative ferromagnetic quantum critical point (QCP) at $P_c approx 1.7 pm 0.02$ GPa. In the ordered phase, ferromagnetic magnons scatter the conduction electrons and give rise to a well defined power law temperature dependence in the resistivity. The coefficient of this term is related to the spin-wave stiffness and measurements of the very low temperature resistivity allow to accompany the behavior of this quantity as the the ferromagnetic QCP is approached. We find that the spin-wave stiffness decreases with increasing pressure implying that the transition to the non-magnetic Fermi liquid state is driven by the softening of the magnons. The observed quantum critical behavior of the magnetic stiffness is consistent with the influence of disorder in our system. At quantum criticality ($P = P_c approx 1.7 pm 0.02$ GPa), the resistivity shows the behavior expected for an itinerant metallic system near a ferromagnetic QCP.
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