  The lepton flavour violating signals $e^{+}e^{-}\to \ell^{+}e^{-}$ and
$e^{-}e^{-}\to \ell^{-}e^- (\ell=\mu,\tau)$ are studied in the context of low
energy R-parity conserving supersymmetry at center of mass energies of interest
for the next generation of linear colliders. Loop level amplitudes receive
contributions from electroweak penguin and box diagrams involving sleptons and
gauginos. Lepton flavour violation is due to off diagonal elements in $SU(2)_L$
doublet slepton mass matrix. These masses are treated as model independent free
phenomenological parameters in order to discover regions in parameter space
where the signal cross section may be observable. The results are compared with
(a) the experimental bounds from the non-observation of rare radiative lepton
decays $\mu, \tau \to e\gamma$ and (b) the general mSUGRA theoretical scenario
with seesaw mechanism where off diagonal slepton matrix entries are generated
by renormalization group evolution of neutrino Yukawa couplings induced by the
presence of new energy scales set by the heavy $SU(2)_L$ singlet neutrino
masses. It is found that in $e^- e^-$ collisions the ($e\tau$) signal can be
observable with a total integrated luminosity of 100 fb$^{-1}$ and the that the
background can be easily suppressed. In $e^+ e^-$ collisions the cross section
is smaller and higher luminosities are needed. The experimental bound on the
decay $\mu \to e \gamma$ prevents the ($e\mu$) signal from being observable.
