Dunham's dramedy Girls, which began airing on US television in April 2012, follows a group of 20-something women in New York City and is partially based on Dunham's own experiences.
Dunham, 26, has already been published in the New York Times and The New Yorker. She also wrote, directed and starred in the 2010 film Tiny Furniture.
In 2008, comedian Sarah Silverman saw the rights for her book The Bedwetter go to HarperCollins for $2.5m, while in the same year Tina Fey won a $6m advance for Bossypants.