  We perform calculations for the production of $\phi$ mesons in nuclei at
energies just above threshold and study the $A$ dependence of the cross
section. We use results for the $\phi$ selfenergy in the medium obtained within
a chiral unitary approach. We find a strong $A$ dependence which is tied to the
distortion of the incident proton and to the absorption of the $\phi$ in its
way out of the nucleus. The effect of this latter process reduces the cross
section in about a factor two in heavy nuclei proving that the $A$ dependence
of the cross section bears valuable information on the $\phi$ width in the
nuclear medium. Calculations are done for energies which are accessible in an
experimental facility like COSY.
