The 29-year-old took on the post from Mark Lamarr in 2006 and has since won a number of awards for his work at the helm.
However, a source told The Sun that Amstell now wants to concentrate on other projects.
"Simon is a funny guy who can think on his feet but he has decided it is time to move on," the insider said. "He is working on a new project that has been commissioned by the BBC and feels he wants to concentrate on that."
A spokeswoman for the corporation confirmed that Amstell would leave the show at the end of his third series.