The Bernstein analyst Ronny Gal said the conversations going forward would most likely have to do with giving Shire shareholders a deal involving more cash; its latest offer contained about 60% in Takeda stock.
Midday Thursday, Allergan confirmed it's in the "early stages of considering a possible offer for Shire", according to a prepared statement.
That offer was rejected by Shire's board as directors said it "significantly undervalued the company, its growth prospects and pipeline".
In addition to making medications like Adderall and Vyvanse, Shire focuses on rare diseases like the blood disorder hemophilia.
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.
Takeda said it would remain disciplined in its approach and meant to maintain its dividend policy and investment grade credit rating, adding that: "Discussions between the parties regarding a potential offer are ongoing".More news: Sean Hannity Is Named as Michael Cohen's Client
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Regardless of the outcome of the talks, Takeda said it remains "well-positioned to continue its transformation, delivering portfolio growth while strengthening its pipeline and boosting profitability".
Shire also rejected the third proposal, made up of £17.75 in cash, to be paid in USA dollars, and £28.75 of new Takeda shares. However, both firms said talks were continuing.
The two earlier cash-and share offers were worth 44 and 45.50 pounds per share respectively.
It is not clear whether Allergan has submitted an offer, and there is no certainty of a bid, the sources said. Now that Allergan has admitted its interest, it has until May 17 to do the same. Shire was up 8.4 per cent to £40.68.
The second bid on April 11 valued Shire at £43 billion, with shareholders in the UK-based firm owning 51% of the enlarged Takeda.
"We think that Takeda's recently expressed interest in Shire may flush out other potential bidders, most likely among the USA pharma heavyweights including Pfizer, Amgen and AbbVie", analysts at United First Partners said in a note, as quoted by Bloomberg.