The fact that Obama once ate dog meat quickly became a subject of both mirth and insult, especially on Twitter, where another campaign-related "dog war" has taken place over the 1983 story of the Romney family driving cross-country with their dog in a carrier tied to the roof of the car.

"What are we going to do, go back to kindergarten and before that and that's going to be part of this debate? It's silly," Axelrod said.

Axelrod has taken a shot at Romney over his dog story in the past, drawing a contrast between Obama and other pet owners by tweeting a photo of the president with his dog, Bo, riding inside the car.

In retaliation, top Romney adviser Eric Fehrnstrom recently tweeted a response to Axelrod's original tweet, adding the line: "In hindsight, a chilling photo."

Still, asked if Twitter is a new battlefront for campaigns in 2012, Axelrod dismissed the idea. 

"I think Twitter can be a tool for getting information out that's valuable," he said. "I think it can be a positive thing." 

Axelrod is well-known for responding via his Twitter feed to topics of the day, and for trading jabs with Fehrnstrom.