Florian Phillippot, the National Front's vice president, said the party could not accept "this insulting connection".
"Marine Le Pen is defending her honour, but also that of party members and supporters and the millions of Front National voters."
Ms le Pen, who took over the party from her father, has fought to move the National Front away from its allegedly extremist and antisemitic past. She came third in the French presidential elections earlier this year.
Madonna is set to perform in Nice next month.
