Top U.S. retailers including Wal-Mart Stores Inc. and Target Corp. are battling to limit a new federal law that could force them to report whether their store-brand goods contain minerals from war-torn Central Africa.

The requirement, part of the Dodd-Frank financial law passed in July, aims to pressure companies to spurn so-called conflict minerals—tin, tantalum, tungsten or gold from parts of the Democratic Republic of Congo or neighboring countries.

Income...