Domino's Pizza CEO Richard Allison said Friday that he worries about the "unintended consequences" that President Donald Trump's trade war with China could have on the company's international franchisees and dairy farmers.

Allison, speaking in an exclusive interview with CNBC's , said the U.S. pizza chain has been able to avoid some of the effects of the tariffs due to the company's practice of sourcing products locally in the U.S. and in many other markets.

There could, however, be some implications for franchisees in international markets that buy products from the U.S., he said.

The Trump administration is attacking what it sees as unfair trade on a number of fronts. A new round of U.S. tariffs on $16 billion worth of Chinese imports kicked in last month, prompting an equivalent retaliation from Beijing.

"Then also we have great partners in the diary farmers in the U.S.," Allison said in the interview Friday, "and we do worry about some of the unintended consequences for them along the way."

American farmers are among the hardest hit by China's retaliatory duties on everything from meats and dairy products to fruits and vegetables to rice and soybeans. The Trump administration unveiled a plan in July to offer billions in aid to farmers hit by tariffs, which was quickly condemned by some Republican senators.

The president is also hitting China and other countries with steel and aluminum tariffs. When asked about that, Allison said he expects a "modest" impact.

Allison spoke to CNBC from the company's largest supply chain facility in Edison, New Jersey. Domino's will have its grand opening for the warehouse on Friday morning. Domino's said it expects the new supply chain to create 150 jobs and supply more than 450 stores.

Domino's announced earlier this year that its then-chief executive, Patrick Doyle, would leave the pizza chain in June and would be succeeded by Allison, who was president of the chain's international business.

The company launched a new initiative this year called Domino's Hotspots, which lets customers order food to places without traditional addresses.

Domino's, with a market cap of $11.7 billion, has seen its share rise more than 48 percent so far this year.

By contrast, its rival, Papa John's, which has a market cap of $1.5 billion, has seen its shares fall 17 percent over the same time period as it deals with the fallout from a controversy involving its founder, John Schnatter, and weak sales. Schnatter resigned from his post as chairman on July 11 after Forbes published an article detailing a May conference call in which Schnatter used the N-word. Schnatter later apologized.

In the interview Friday, Allison said Papa John's current problems reminded the company to focus on its customers and franchisee.

Allison also commented on the NFL, which has suffered from poor ratings. He said he doesn't expect any impact on the company's earnings.