Shake Shack is considering the launch of a Chick-fil-A rival, according to a company filing.

The burger chain's subsidiary, SSE IP, filed an application to trademark the name "chicken shack," CNBC reports.

That could mean the company is testing a chicken sandwich or a new chain with a chicken-based menu.

Shake Shack hasn't revealed any details of its plans.

"Shake Shack was born from a fine dining company, and we constantly test new menu items in our test kitchen," the company told CNBC in a statement. "We have no new items to announce at this time, except for the new ParkBurger at New York's Madison Square Park, which reopened today."

By adding chicken to the menu, Shake Shack would be taking direct aim at Chick-fil-A, the leader in the fast-fast food chicken market. The popular Atlanta-based chain surpassed KFC last year to become the No. 1 chicken chain in the US.

View photos julian marchese, chick-fil-a, chicken sandwich, lunch More

The menus at Shake Shack and Chick-fil-A are already pretty similar. Both feature sandwiches, french fries and milkshakes. But the only menu item containing chicken currently at Shake Shack is the chicken dog, which is a hot dog made with chicken, apple and sage sausage.

Shake Shack wouldn't be the first chain to go after Chick-fil-A's market.

Taco Bell parent company Yum! Brands has also been testing a chicken sandwich chain, called Super Chix.

View photos Super Chix More

Like Chick-fil-A, Super Chix sells fried chicken sandwiches, chicken strips, frozen custard, and french fries.

According to research firm NPD Group, chicken meals rose 3% to $5.4 billion for the year ending in March at fast-food restaurants, CNBC noted.

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