We perform calculations for the production of φ mesons in nuclei at energies just above threshold and study the A dependence of the cross section. We use results for the φ self-energy 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 φ on its way out of the nucleus. The effect of this latter process reduces the cross section by about a factor of 2 in heavy nuclei, proving that the A dependence of the cross section bears valuable information on the φ width in the nuclear medium. Calculations are done for energies accessible in an experimental facility like COSY