  In this study, we demonstrate the potential of the NA64 experiment at CERN
SPS to search for New Physics processes involving $e\rightarrow\mu$ transitions
after the collision of 100 GeV electrons with target nuclei. A new Dark Sector
leptonic portal in which a scalar boson $\varphi$ could be produced in the
lepton-flavor-changing bremsstrahlung-like reaction, $eN\rightarrow \mu
N\varphi$, is used as benchmark process. In this work, we develop a realistic
Monte Carlo simulation of the NA64 experimental setup implementing the
differential and total production cross-section computed at exact tree-level
and applying the Weisz\"{a}cker-Williams phase space approximation. Using this
framework, we investigate the main background sources and calculate the
expected sensitivity of the experiment. The results indicate that with minor
setup optimization, NA64 can probe a large fraction of the available parameter
space compatible with the muon $g-2$ anomaly and the Dark Matter relic
predictions in the context of a new Dark Sector leptonic portal with $10^{11}$
EOT. This result paves the way to the exploration of lepton-flavour-changing
transitions in NA64.
