MONTERREY, Mexico—American trade negotiators are taking aggressive measures to close what they describe as a Mexican “back door” through which steel and other auto-manufacturing components produced outside North America are sold tariff-free in the U.S.

The U.S. trade representative has proposed a significant expansion of the North American Free Trade Agreement so-called tracing list for auto manufacturing, which enumerates parts that require stringent proof of where they were originally made, according to three people familiar...