"Proposals to ban menthol cigarettes in the USA have been for the table for many years and we will continue to engage with the FDA on the topic", said spokesman of Imperial Brands.
BAT owns Newport, the leading menthol brand in the U.S., and sales of the mint-flavoured products there are estimated to bring in a quarter of the company's revenues, according to analysts at Morgan Stanley.
The US Food and Drug Administration will this week impose a ban on the sale of most flavoured e-cigarettes in tens of thousands of convenience stores and petrol stations across the US.
E-cigarette flavors such as mint, mango, and cucumber are much more appealing to teens than the harsh tobacco flavor of regular cigarettes - making it immensely easier for young people to get hooked on the addictive chemical.
Given the FDA's position that it will follow a science-based approach, it is unlikely the regulatory body will outright ban menthol - although a proposal to restrict or lower the levels of menthol is reasonable, the analyst said.
The potential menthol ban won't have a near-term effect on the markets, according to analysts.
Earlier this year, San Francisco passed a prohibition against the sales of menthol cigarettes and flavored e-cigarettes.
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FDA plans to ban menthol cigarettes and flavored e-cigarettes.
The decline in BAT shares illustrates that while the Lucky Strike maker is expanding into vapor and heated-tobacco products, its main business is still cigarettes. Traditional products make up most of its revenue.
Approximately 21 percent of Altria's volume is menthol and it accounts for around 18 percent of total profit, while about 20 percent of Imperial Tobacco's volume is menthol, the analyst said.
When BAT stepped up its presence in the United States past year by acquiring the 58% of Reynolds it did not already own for $50bn, it listed its menthol cigarettes as a key attraction.
If the ban goes through, it would be a win for public health advocates who say a large number of African Americans have become addicted to menthol cigarettes.
BAT shares have dropped 40% this year, the biggest drop in at least two decades.