Shire said in a statement that the cash-and-stock offer arrived May 30 and was rejected by the board of directors because it was "highly conditional" and undervalued Shire.
AbbVie said in a statement that it had made three offers to Shire and, though talks have stopped for now, might make a fourth offer in the future.
Shire is listed on the London Stock Exchange, and its American depository receipts (ADR) trade on the Nasdaq market. The AbbVie offer represented a 23 percent premium on Shire's Thursday closing price of $191.71 on Nasdaq and 37.40 British pounds on the LSE. Friday on Nasdaq, Shire rose $31.18, or 16.3 percent, to close at $222.89.
"This sounds like a fair offer, and we are surprised that Shire does not seem interested in engaging with AbbVie," Bernstein Research analyst Ronny Gal wrote in a note to clients. "We believe that if Shire wants more than 46 [pounds] per share (ADR $230) it may be difficult for anyone to pay that."
Gal said U.S.-based Allergan and U.K.-based AstraZeneca might be interested. Allergan is still fighting off a takeover bid by Valeant Pharmaceuticals. AstraZeneca, which has operations in Delaware, fought off a takeover bid by Pfizer, which also wanted its tax domicile to be in the United Kingdom.
Nearly 41 percent of Shire's revenue comes from three drugs approved to treat attention deficit hyperactivity disorder, led by Vyvanse.
Susan Kilsby, chairwoman of Shire's board, released a statement after the rejection: "The board believes the proposal fundamentally undervalued Shire and its prospects and that as an independent company Shire's focused growth strategy will continue to deliver significant shareholder value and patient benefits."