The days of passengers bringing their pets on airplanes as emotional-support animals could be ending.

The U.S. Department of Transportation on Wednesday proposed that only specially trained dogs qualify as service animals, which must be allowed in the cabin at no charge.

Airlines could ban emotional-support animals including untrained dogs, cats and more exotic companions such as pigs, pheasants, rabbits and snakes.

Airlines say the number of support animals has grown dramatically in recent years. They lobbied the Transportation Department to crack down on what they consider a scam — passenger...