As the owner of a very pettable dog* in a French speaking community, what I hear most often is

"puis-je le caresser?" or
"je peux le caresser?"

The nickname for dogs, FYI, is "toutou", so if you say to someone "puis-je caresser le toutou?" you will probably get an affirmative response. But always ask first; some dogs aren't nearly as friendly as they look. If you are allowed to pet the dog, it's customary to make some complimentary noises afterwards, like
"comme il est si beau!" (or mignon or charmant or gentil)

The nickname for cats is "nounou". We were driving through Alsace half listening to the radio when suddenly the DJ shouted "oh les pauvres petits nounous!" in a mock tragic tone and it has become a catch phrase around the house.

*Our dog is an English cocker who is a dead ringer for the cocker spaniel Spot in the old Fun with Dick and Jane books. Although she *looks* very cuddly, she doesn't really like strangers. Fortunately, she's too well bred with the innate English cocker good manners to cause any problems with unwanted "caresses."

I must add that you should NOT pet a strange dog by patting it on the head! That can be interpreted by the dog as an aggressive gesture (or so I've been told by several professional dog trainers and breeders). The best thing to do is let the dog sniff you hand, and if things are ok, then scratch it gently on the front of the chest.

BTilke,

Merci for the information!