The European Union's battle over London's financial services sector risks dragging an unexpected player into the fray: Uncle Sam.

That's what could happen if French or German leaders attempt to pry away the some US$533 trillion of swaps cleared by London Stock Exchange (LSE) Group Plc's LCH unit annually. In a sign of how badly the EU wants London's financial turf post-Brexit, talks are already underway within the EU to limit euro clearing outside their jurisdiction, a move that would affect the U.S. too, LSE chief Xavier Rolet said Wednesday.

Rolet has said New York, where LCH also holds a clearing license, is the only financial center that could practically clear interest-rate swaps. A move to the U.S. financial hub would potentially pit leaders like French President Francois Hollande against peers in Washington—meaning the EU would be in a battle against the world's largest economy as it seeks to negotiate a new relationship with the U.K.