I don't know about your cats but mine have shown no interest in wet laundry, or dirty laundry, only fresh warm dry laundry. But since my building's laundry room is in the basement I don't really have experience with having a machine in the house.

When I had kids I would fold the laundry down there, for a few quiet moments of feeling organized. Now I bring it up immediately and chirrup as I dump it on the spare bed so the cats can enjoy it first.