Jameson was born in Leith on 11 July 1774, the son of Catherine Paton (1750–94) and Thomas Jameson (c.1750–1802), a soap manufacturer on Rotten Row (now Water Street). They lived on Sherrif Brae. His early education was spent at Leith Grammar School, after which he became the apprentice of the Leith surgeon John Cheyne (father of John Cheyne), with the aim of going to sea. He also attended classes at the University of Edinburgh (1792–93), studying medicine, botany, chemistry, and natural history. His father's brother Robert Jameson, was also a physician and lived with them on Rotten Row.