Japan's Takeda Pharmaceutical has increased its offer for Shire to $47 a share after its three previous bids - the last at £46.50 - were rejected by the rare-disease drugs specialist.
The offer, comprising £21 in cash and £26 in new Takeda shares, values London-listed Shire at £44.3 billion.
This represents a 7% increase on its first offer and a 58% premium on Shire's undisturbed share price, Takeda said.
Takeda said it hoped to engage with the board of Shire and agree to extend a deadline imposed by the British Takeover Panel of April 25.
Shire, which also makes drugs to treat attention deficit hyperactivity disorder (ADHD), rejected the previous three offers, saying they significantly undervalued the company, its growth prospects and its pipeline.
Buying Shire would be the largest ever overseas acquisition by a Japanese company and propel Takeda, led by Frenchman Christophe Weber, into the top ranks of global drugmakers.
It would be Weber's boldest move by far, significantly boosting Takeda's position in rare diseases, including a blockbuster haemophilia franchise, gastrointestinal disorders and neuroscience, where Shire is a leader in ADHD drugs.
But it would be a big financial stretch since Shire, with a market value of more than £34 billion, is worth a lot more than Japan's biggest drugmaker, which has a market capitalisation of 4.1 trillion yen.
Botox maker Allergan had said yesterday that it was considering an offer for Shire before deciding otherwise.