Kroger announced Monday that a California subsidiary will no longer accept Visa credit cards as of mid-August as the two companies clash over payment fees and rates.

"Visa's rates and fees are among the highest of any credit card brand," Foods Co. said in an emailed statement Monday. "The savings will be passed along to Foods Co. customers in the form of low everyday prices on the items shoppers purchase most."

The ban on Visa credit cards will take effect starting Aug. 14 and affect 21 Foods Co. stores and fuel centers in central and north California. Though stores in major cities like San Francisco and Sacramento will be affected by the change, customers will still be able to use Visa debit cards, MasterCard, American Express and Discover credit cards as usual, the statement added.

"There are ongoing discussions about payment technologies," Christopher Hjelm, executive vice president and chief information officer at Kroger, told CNBC. "We've been working to get the economics right and we are taking a step to make sure we continue to get our customers the best value."

Visa said it was frustrated with Kroger's move. "Visa is disappointed at Kroger's decision to stop accepting Visa credit cards at its Foods Co. stores," it said in a statement. "When consumer choice is limited, nobody wins. Our goal is to protect the interests of our cardholders to ensure they can use their Visa credit cards wherever they shop. Visa remains committed to working with Kroger to reach a reasonable solution."

Shares of Visa fell 3 percent Monday amid the news while Kroger's stock closed up 3.5 percent.

The dispute between Kroger and Visa is just the latest in a string of battles over the tens of billions of dollars merchants pay each year in swipe fees.

For example, the Supreme Court concluded earlier this year that American Express rules that don't allow merchants to offer promotions or discounts on rival cards that charge them lower fees don't violate federal antitrust law.

The Justice Department initially sued Amex and its larger rivals Mastercard and Visa, arguing that merchants had no real bargaining power when it came to using their payment services. A lower court had ruled that Amex's "anti-steering" rules stifled competition.