  We use a functional renormalization group (fRG) approach to investigate
potential interaction-induced instabilities in a two-dimensional model for the
Dirac nodal-line materials ZrSiS and ZrSiSe employing model parameters derived
from {\it ab initio} calculations. Our results confirm that the excitonic
instability recently found in random-phase approximation for ZrSiS is indeed
the leading instability. In the simplest modeling, spin- and charge-excitonic
states are degenerate. Beyond this, we show that the fRG analysis produces an
energy scale for the onset of the instability in good agreement with the
experimentally observed mass enhancement. Additionally, by exploring the
parameter space of the model we find that reducing the band splitting increases
the instability scale and gives the chance to drive the system into an
unconventional superconducting pairing state. The model parameters for the case
of the structurally similar material ZrSiSe suggest the $d$-wave
superconducting state as the leading instability with a very small critical
scale.
