  Motivated by the many associations of $\gamma$-ray bursts (GRBs) with
energetic supernova (SN) explosions, we study the propagation of relativistic
jets within the progenitor star in which a SN shock wave may be launched
briefly before the jets start to propagate. Based on analytic considerations
and verified with an extensive set of 2D axisymmetric relativistic hydrodynamic
simulations, we have estimated a threshold intrinsic jet luminosity, $L_{\rm
j}^{\rm thr}$, for successfully launching a jet. This threshold depends on the
structure of the progenitor and, thus, it is sensible to its mass and to its
metallicity. For a prototype host of cosmological long GRBs, a low-metallicity
star of 35$\,M_{\odot}$, it is $L_{\rm j}^{\rm thr} \simeq 1.35\times
10^{49}\,$erg$\,$s$^{-1}$. The observed equivalent isotropic $\gamma$-ray
luminosity, $L_{\rm \gamma,iso,BO} \simeq 4 \epsilon_\gamma L_{\rm j}
\theta_{\rm BO}^{-2}$, crucially depends on the jet opening angle after
breakout, $\theta_{\rm BO}$, and on the efficiency for converting the intrinsic
jet luminosity into $\gamma$-radiation, $\epsilon_\gamma$. Highly energetic
jets can produce low-luminosity events if either their opening angle after the
breakout is large, which is found in our models, or if the conversion
efficiency of kinetic and internal energy into radiation is low enough. Beyond
this theoretical analysis, we show how the presence of a SN shock wave may
reduce this luminosity threshold by means of numerical simulations. We foresee
that the high-energy transients released by jets produced near the luminosity
threshold will be more similar to llGRBs or XRFs than to GRBs.
