  In this article we investigate the prospects of searching for sterile
neutrinos in lowscale seesaw scenarios via the lepton flavour violating (but
lepton number conserving) dilepton dijet signature. In our study, we focus on
the final state $e^\pm \mu^\mp jj$ at the HL-LHC and the FCC-hh (or the SppC).
We perform a multivariate analysis at the detector level including the dominant
SM backgrounds from di-top, di-boson, and tri-boson. Under the assumption of
the active-sterile neutrino mixings $|V_{ l N}|^2=|\theta_e|^2=|\theta_\mu|^2$
and $|V_{ \tau N}|^2 = |\theta_\tau|^2=0$, the sensitivities on the signal
production cross section times branching ratio $\sigma(p p \to l^\pm N)\times
{\rm BR} (N \to l^{ \mp} jj)$ and on $|V_{ l N}|^2$ for sterile neutrino mass
$M_N$ between 200 and 1000 GeV are derived. For the benchmark $M_N=500$ GeV,
when ignoring systematic uncertainties at the HL-LHC (FCC-hh/SppC) with 3 (20)
${\rm ab}^{-1}$ luminosity, the resulting 2-$\sigma$ limits on $|V_{ l N}|^2$
are $4.9\times 10^{-3}$ ($7.0\times 10^{-5}$), while the 2 -$\sigma$ limit on
$\sigma \times {\rm BR}$ are $4.4\times10^{-2}$ ($1.6\times10^{-2}$) fb,
respectively. The effect of the systematic uncertainty is also studied and
found to be important for sterile neutrinos with smaller masses. We also
comment on searches with $\tau^\pm \mu^\mp jj$ and $\tau^\pm e^\mp jj$ final
states.
