We use Chandra X-ray data and Very Large Array radio observations for a sample of 20 nearby, massive, X-ray bright, early-type galaxies to investigate the relation between the Bondi accretion rates and the mechanical jet powers. We find a strong correlation ($rho = 0.96^{+0.03}_{-0.09}$; BF10 > 100) between the Bondi accretion power, PBondi, and the mechanical jet power, Pjet, for a subsample of 14 galaxies, which also host cool H α+[N II] line emitting gas and thus likely have thermally unstable atmospheres. The relation between the Bondi accretion power and the mechanical jet power for this subsample is well described by a power-law model $log frac{P_{mathrm{Bondi}}}{{10^{43} , mathrm{erg , s^{-1}}}} = alpha + beta log frac{P_{mathrm{jet}}}{{10^{43} , mathrm{erg , s^{-1}}}}$, where α = 1.10 ± 0.25 and β = 1.10 ± 0.24 with an intrinsic scatter $sigma = 0.08^{+0.14}_{-0.06}$ dex. The results indicate that in all galaxies with thermally unstable atmospheres the cooling atmospheric gas feeds the central black holes at a similar jet-to-Bondi power ratio. For the full sample of 20 galaxies, the correlation is weaker and in a subset of galaxies with no signs of H α+[N II] emission, we see a hint for a systematically lower jet-to-Bondi power ratio. We also investigate the dependence of jet power on individual quantities in the Bondi formula such as the supermassive black hole mass (M∙) and the specific entropy of the gas (K) at the Bondi radius. For the subsample of H α+[N II] emitting galaxies, we find a very tight correlation of Pjet with M∙ ($rho = 0.91^{+0.06}_{-0.11}$; BF10 > 100) and, although poorly constrained, a hint of an anticorrelation for Pjet and K ($rho = -0.47^{+0.60}_{-0.37}$; BF10 = 1.1).
