“This proposal is the culmination of a number of prior discussions between Mylan and Perrigo about the compelling strategic and financial logic of this combination,” Mr. Coury said in a statement. “This combination would result in meaningful immediate and long-term value creation, and our proposal is designed to deliver that value to shareholders and other stakeholders of both companies. We have great respect for Perrigo’s board and management team and what they have built.”

Perrigo acknowledged it had received the offer and said it would respond after its board had met. But it is unlikely that the company will simply accept Mylan’s offer of $205 a share. Instead, it may deem the offer to be too low in the hope of attracting a higher bid. Or it could enter into friendly negotiations with Perrigo, and perhaps other potential buyers.

Image Mylan and Perrigo would have more than $15 billion in combined annual sales. Credit... Travis Dove for The New York Times

Manufacturers of generic drugs, like Mylan, have increasingly sought to diversify their portfolios as revenues for their product lines have slowed in recent years, said Michael Faerm, an analyst for Wells Fargo. Fewer brand-name drugs have been losing patent protection, and competition from new players — especially manufacturers in India — has grown more intense.

“So the garden-variety generics in many cases — that opportunity is not what it once was,” Mr. Faerm said in an interview. “In order to keep growing, they need to look elsewhere.”

Some companies, like Teva Pharmaceutical and Actavis, have sought to expand into brand-name drugs, while others, including Mylan, have focused on harder-to-make generic drugs like respiratory products and extended-release medications. That is one reason Perrigo, with its portfolio of dermatologic creams and over-the-counter drugs, looks attractive to Mylan, he said.

“This could be a strong deal” for Mylan, Mr. Faerm wrote in a note. “It would bring together highly complementary strengths at the two companies.”