  We discuss the dependence of J/\psi\ production on the charged particle
multiplicity in proton-proton collisions at LHC energies. We show that, in the
framework of parton saturation or string interaction models, the hard J/\psi\
production exhibits a significant growth with the multiplicity, which is
stronger than linear in the high density domain. This departure from linearity,
that should affect any hard observable, applies for high multiplicity
proton-proton collisions in the central rapidity region and is a consequence of
the parton saturation or the strong interaction among colour ropes that take
place at LHC energies. Our assumption, the existence of coherence effects
present in proton-proton collisions at high energy, can also be checked by
studying the particular shape of the probability distribution associated to the
J/\psi\ production.
