The same day Bristol-Myers Squibb Co. shook up its board to satisfy one activist investor, the drugmaker was faced with another: Carl Icahn, whose interest is fueling speculation the company could soon be put on the auction block.

The appearance of Mr. Icahn, an investor with a history of pushing for deals among pharmaceutical companies, surprised Bristol-Myers executives, who had just avoided a potential fight with activist hedge fund Jana Partners LLC and are still reeling from a costly stumble in the company’s effort to...