So, a little "hard-grated cheese" for your pasta? It doesn't have quite the same ring as Parmesan.
U.S. dairy producers, cheesemakers and food companies are all fighting the idea, which they say would hurt the $4 billion domestic cheese industry and endlessly confuse consumers.
"It's really stunning that the Europeans are trying to claw back products made popular in other countries," says Jim Mulhern, president of the National Milk Producers Federation, which represents U.S. dairy farmers.
The European Union would not say exactly what it is proposing or even whether it will be discussed this week as a new round of talks on an EU-United States free trade agreement opens in Brussels.
European Commission spokesman Roger Waite would only say that the question "is an important issue for the EU."
That's clear from recent agreements with Canada and Central America, where certain cheese names were restricted unless the cheese came from Europe. Under the Canadian agreement, for example, new feta products manufactured in Canada can only be marketed as feta-like or feta-style, and they can't use Greek letters or other symbols that evoke Greece.
Though they have not laid out a public proposal, the EU is expected to make similar attempts to restrict marketing of U.S.-made cheeses, possibly including Parmesan, Asiago, Gorgonzola, feta, fontina, grana, Muenster, Neufchatel and Romano.
And it may not be just cheese. Other products could include bologna, Black Forest ham, Greek yogurt, Valencia oranges and prosciutto, among other foods.
The trade negotiations are important for the EU as Europe has tried to protect its share of agricultural exports and pull itself out of recession. The ability to exclusively sell some of the continent's most famous and traditional products would prevent others from cutting into those markets.
Concerned about the possible impact of changing the label on those popular foods, a bipartisan group of 55 senators wrote U.S. Trade Representative Michael Froman and Agriculture Secretary Tom Vilsack this week asking them not to agree to any such proposals by the EU.
Led by New York Sen. Charles Schumer, D-N.Y., and Pennsylvania Sen. Patrick Toomey, R-Pa., the members wrote that in the states they represent, "many small- or medium-sized, family owned businesses could have their businesses unfairly restricted" and that export businesses could be gravely hurt.
Schumer said artisanal cheese production is a growing industry across New York.
"Muenster is Muenster, no matter how you slice it," he said.
Trevor Kinkaid, a spokesman for the U.S. trade representative, said conversations on the issue are in the early stages but that the U.S. and E.U. have "different points of view" on the topic.
The agency wouldn't disclose details of the negotiations, but Kinkaid said the U.S. government is "committed to increasing opportunity for U.S. businesses, farmers and workers through trade."