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

Struggling cosmetics seller Avon Products Inc. on Monday rejected a $10 billion buyout offer from Coty Inc., 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. But investors frustrated with the company’s shrinking profits and disappointing sales growth sent the stock soaring 17 percent on the news.

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.

New York-based Coty is mainly known in the U.S. for its Calvin Klein fragrance lines and perfumes sold under the names of celebrities including Beyonce, Lady Gaga and Celine Dion. Coty went public with its bid after private overtures failed to result in negotiations, but said it won’t pursue a hostile takeover.

Founded in Paris in 1904 by perfume pioneer Francois Coty, the company sells products in 135 markets across the globe and had revenue of $4.1 billion last year. It has said it wants to increase that to $7 billion by 2015. Adding Avon’s $11 billion a year in revenue would far surpass that goal. The deal also would strengthen Coty’s hand in emerging markets.

In the past two years Coty has snapped up smaller beauty brands such as OPI nail polish and skin care products maker Philosophy Inc.