The activist fund manager has acquired what was described as a significant stake, according to the Financial Times, at a time when concerns about its performance and that of chief executive Emma Walmsley have been growing said the newspaper.
Elliot is renowned for taking stakes and forcing changes on management and the structure of businesses, perhaps most famously at Whitbread, where it called for the splitting away of coffee chain Costa and at BHP where it wanted the miner to divest its US petrol assets and end its dual-listed share structure.
A partnership with French peer Sanofi has disappointed, with their candidate having to be reformulated after trials. A project with the Chinese group Clover has been ended.
Next year, the group is separating the consumer health business from its pharma and vaccine division, but this has not helped the share price, which has fallen more than 10% since Walmsley was appointed in April 2017.
Astra, by contrast, has risen by 50% over the same time rame.
Elliott and GSK, which has a market value of GBP65bn, both declined to comment said the FT.
Glaxo shares rose 6% to 1,365p.