The hedge fund owns a stake of more than 10% which is valued at more than $800 million, the person said, declining to reveal the exact size of the investment.
Dropbox, which was co-founded by Drew Houston who is currently the company's chief executive officer, is valued at roughly $11 billion. The company, which allows users to store documents, videos and photos online, listed its shares in March 2018 at $21 a share.
A representative for Dropbox did not immediately respond to a request for comment.
On Wednesday Dropbox was trading at $28.11, having jumped as much as 7% on news that Elliott has a stake and is holding discussions with management.
The Wall Street Journal first reported the stake.
Elliott has previously worked with companies that were being lead by founders, including Twitter and SoftBank Group.
The hedge fund invests roughly $40 billion and currently owns stakes in Duke Energy, where it has urged the company to consider splitting into three companies, and drugmaker GlaxoSmithKline.
(Reporting by Akanksha Rana and Svea Herbst-Bayliss; Editing by Aditya Soni and David Gregorio)