NEW YORK — Coty came calling, but Avon slammed the door.

Struggling cosmetics seller Avon Products on Monday rejected a $10 billion buyout offer from Coty, a smaller beauty products maker looking to capitalize on Avon’s business woes. The $23.25-per-share bid marked a 20 percent premium to Avon’s closing stock price Friday. Avon said that was too low.

It’s been a steep slide for an American icon. Founded in 1886, Avon became a fixture in households across the country as its legions of “Avon ladies” went door to door selling makeup to family, friends and acquaintances. But North American sales have long been in decline. Now, about 80 percent of Avon’s $11 billion in annual revenue comes from overseas.

The company’s profit has shrunk over the past three years. It has frequently missed analysts’ earnings expectations and posted weak sales in some of its largest markets, including Brazil and Russia. Avon also faces a bribery probe that started in China and widened to other countries. The Securities and Exchange Commission is investigating Avon’s contact with financial analysts in 2010 and 2011 related to the investigation.

Investors and analysts have blamed CEO Andrea Jung for being slow to react to declining results and wrap up the bribery investigation. In December, Avon began seeking a replacement for the 12-year veteran, who will remain chairman.