Gov. Scott Walker of Wisconsin and his wife, Tonette, had planned to spend a recent Friday night at the Nashua home of Jennifer Horn, the New Hampshire Republican chairwoman.

Then he heard about Al, the Horns’ beloved Dalmatian-Catahoula Leopard mix.

The dinner was promptly moved to a restaurant in nearby Bedford. “The governor’s allergic to dogs,” Ms. Horn said. “And we have a very hairy dog.”

The attention to Mr. Walker’s likely candidacy for the Republican presidential nomination has focused on weighty matters such as his battles with the left, faltering forays into foreign policy and conservative stances on social issues including abortion and gun rights. But little notice has been given to an area in which he faces a different sort of constitutional challenge: overcoming his aversion to man’s best friend.

Image In 2010, during his initial run for governor of Wisconsin, Scott Walker greeted Lisa Bell, a Republican activist who had been tossing her Wire Fox terrier, Diva, in the air during his speech. Credit... Mark Hertzberg/Journal Times, via Associated Press

Jeb Bush can lament how he lost a Labrador (named for his brother Marvin) to cancer. Marco Rubio has a Shih Tzu, with a name like a gift from heaven: Manna. Ted Cruz goes one better: His rescue mutt is called Snowflake. (“Dear Jesus, please, please, PLEASE bring us a puppy,” his daughters prayed, according to Mr. Cruz’s Facebook page.) And if Mr. Walker makes it to November, he could face Hillary Rodham Clinton and her toy poodle, Tally.