It was the second time in less than a week that Mylan raised its bid, this time to USD 35.6 billion, to merge the two into a company with USd 15 billion in annual sales.
And it was the second time Perrigo responded within hours with a flat "no," saying the offer deeply undervalued the company.
Even as that happened, Israel's generics giant Teva restated its interest in buying Mylan despite a blistering rejection from Mylan earlier this week.
But analysts say both Mylan and Perrigo are ripe for takeovers in the next five years due to pressures in the global pharmaceutical industry even if they survive the current onslaught.
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"The 'who' and 'when' are extremely unclear," said Michael Waterhouse, an equity analyst at Morningstar. "I would say that eventually, a lot of these companies get bought."
Generic drug companies are under pressure to do deals now because there are fewer big-money drugs shifting to generic status compared with a few years ago, when cholesterol medication Lipitor and other blockbusters were available, Waterhouse said.
Mylan's latest move offered a combination of cash and Mylan shares that amounts to USD 232.23 per Perrigo share, Mylan said.
That was USD 10 a share higher than its April 24 offer, according to Mylan calculations.
Mylan chief executive officer Heather Bresch said a combination with Perrigo would create "a one-of-a-kind global healthcare company" with "complementary businesses and cultures, unmatched scale in its operations and infrastructure, broad and diverse portfolio, and immense reach across distribution channels around the world."
"Today's announcement from Mylan continues to propose a price lower than the previously rejected proposal," Perrigo said.
Mylan has kept up its push even while fending off ostensibly unwanted attention from Teva, which made a USD 40.1 billion offer for it on April 21.
A merger with the Israeli firm would form a USD 28 billion company that Teva said could generate USD 2 billion in annual savings, while expanding its specialty pharmaceutical business.