Here is a link to private day care centers (asili) in Genoa.  (All information on this link is in Italian)

Salita dal Noce
Perasso (close to the Pronto Soccorso of San Martino)
Tel. 010 377 8094
This private school takes over 3 year olds. Children must be potty trained. It is hard to get into this school and there is a long waiting list, but worth a try. Children are separated into age groups during the day. The youngest children have a nap area. No snacks before lunch which may be difficult for the very young ones.

Off Via Berghini
San Fruttuoso
Tel. 010 352 712
A public school which takes children from 2 and ½ years onwards for half days (until noon). They have no area for naps, but they do give a morning snack and help with potty training. From 3 years old children can stay the whole day.

San Martino Hospital
Tel. 010 5551 (switch board)
San Martino Hospital has an asilo nido, for children from a few months old. Hospital employees have priority and a slight discount, but it is open to anyone. They have a nice garden and, although it is technically not permitted under Italian regulations, they will give the babies breast milk. You can get in touch with them through the hospital's operator: 010/5551.

Via Riboli
Tel. 010 362 5348
They have one teacher, Sabrina, who speaks English well in addition to the three owners, who don't really speak English at all, but can understand a little. (February 2014)