Shireconfirmed the news in a separate statement, noting that it had received three conditional proposals from Takeda regarding a possible offer.
A bidding war threatened to break out for Shire after the drugs company rejected a £44 billion offer from a Japanese predator and an Irish rival revealed that it was weighing up an offer.
The offer for the United Kingdom rare diseases specialist is apparently pitched at £46.50 per share, which would value Shire at £42.4bn. Updating the market on Thursday, the Japanese firm revealed Shire's board had rejected its offer, which was equivalent to £46.50 a share. The firm has said it could become acquisitive and is looking at strategic options.
Shire plc, a biotechnology company, focused on serving people with rare diseases and other specialized conditions worldwide.
But it would be a big financial stretch since Shire, with a market value of more than 34 billion pounds ($48.3 billion), is worth a lot more than Japan's biggest drugmaker.
Ahead of Allergan's announcement, shares had plunged on the merger chatter.
Takeda said in a statement that discussions regarding a potential offer are ongoing. The first valued Shire at up to £44.00 per share, the second was £45.60 and the third was £46.50.
The proposed deal comprised £17.75 in cash and £28.75 of new Takeda shares.
However, Shire suggested it would be willing to consider an improved offer: "At the board's request Shire's advisers entered into a dialogue with Takeda's advisers to discuss whether a further, more attractive, proposal may be forthcoming and to understand the basis on which such a proposal would be made".
Based on Takeda's current market capitalisation, Shire shareholders would own approximately 51 per cent. of the enlarged Takeda.
"There can be no certainty an offer will be made nor as to the terms on which such offer would be made", Allergan said.
The Dublin-based biotech specialist said the proposal "significantly undervalued the company, its growth prospects and pipeline".
Shire said in January that it would run its attention deficit hyperactivity disorder (ADHD) business, which consists mainly of its blockbuster drug Vyvanse, separately and possibly seek a separate listing.
This week Shire said it was selling its cancer treatment unit to France's Servier for £1.7bn in a deal that was initiated in December and was unrelated to the possibility of a takeover bid.
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