Medeva's shares added almost 19.5p, or 17 per cent, yesterday, at 135.5p.
Shire's shares closed up 5p at 583.5p. Both companies specialise in treatments for hyperactive children.
Dr Bill Bogie, Medeva's chief executive would not comment on whether the company was in talks. "We are not expecting to make an announcement about the rise in our shareprice [today]," he said.
Sources close to Medeva say that it rejected a cash and paper approach from Andover-based Shire, thought to be in the region of 190p per share and fully underwritten.
Medeva's institutional shareholders are said to have pressured the board to re-enter talks with Shire. Another source said Shire had failed to meet Medeva's demands for a 145p per share offer.
Rolf Stahel, Shire's chief executive, said that the company would be making no announcement today.
"What we have said is that Shire is expanding into Europe and Shire is looking for mergers and acquisitions. Once we have a deal signed, we will make an announcement."
Shire was recently outbid by a US rival in its approach to NeXstar Pharmaceuticals, the US biotechnology group.
There were also suggestions yesterday that Dr Bogie is leading a management buyout for Medeva. Dr Bogie declined to be drawn on the matter.
Medeva's shares have fallen from a five year high of 330p in the last two years as its lead hyperactivity product, methylphenidate, has come up against competition from pharmaceuticals majors.
Analysts say its attraction for Shire is its European marketing base.