Japan’s Takeda Pharmaceutical is in talks with Shire after making a £43bn bid for the London-listed rare diseases drug maker.
Shire’s board rejected the cash and stock offer of £46.50 a share that was made on 12 April.
However, Takeda said that discussions between the two firms about a potential offer were continuing.
Shire confirmed it had opened talks with Takeda to determine if it could make a higher offer.
Shares in Shire rose 4% to £39.15, having risen from about £31 a share a month ago amid takeover speculation.
Late last month Takeda said it was considering making an offer for Shire.
The Japanese firm said it was interested in a takeover to strengthen its cancer, stomach and brain drug portfolio.