The official website for 10 Downing Street hosts a section devoted to ePetitions. One started by a fan of the motoring show calls for Brown to step down and make Clarkson, 47, prime minister instead.
Some 30,000 people have signed the petition, making it the sixth most popular posted on the site.
“We the undersigned petition the Prime Minister to make Jeremy Clarkson prime minister” reads the Clarkson petition.
Clarkson has become popular in Britain for his forthright opinions on everything from immigration to fashion, but so far hasn’t commented on the online campaign.
Some sections of the British media have rallied behind the idea, with the conservative Daily Mail newspaper running a full page of suggested policies Clarkson should adopt if he were to suddenly find himself running the country.