  We derive a sum rule which shows that the Froissart-Martin bound for the
asymptotic behaviour of the $\pi\pi$ total cross sections at high energies, if
modulated by the Lukaszuk-Martin coefficient of the leading $\log^2 s$
behaviour, cannot be an optimal bound in QCD. We next compute the total cross
sections for $\pi^+ \pi^-$, $\pi^{\pm} \pi^0$ and $\pi^0 \pi^0$ scattering
within the framework of the constituent chiral quark model (C$\chi$QM) in the
limit of a large number of colours $\Nc$ and discuss their asymptotic
behaviours. The same $\pi\pi$ cross sections are also discussed within the
general framework of Large-\Nc~QCD and we show that it is possible to make an
Ansatz for the isospin $I=1$ and $I=0$ spectrum which satisfy the
Froissart-Martin bound with coefficients which, contrary to the Lukaszuk-Martin
coefficient, are not singular in the chiral limit and have the correct
Large-\Nc~counting. We finally propose a simple phenomenological model which
matches the low energy behaviours of the $\sigma_{\pi^{\pm}\pi^0}^{\rm
total}(s)$ cross section predicted by the C$\chi$QM with the high energy
behaviour predicted by the Large-$\Nc$ Ansatz. The magnitude of these cross
sections at very high energies is of the order of those observed for the $pp$
and $p{\bar p}$ scattering total cross sections.
