Campaigners say the best friends should marry as a way to encourage tolerance of gay people.
Nearly 7,000 have signed the petition, with more than 3,000 joining a Bert and Ernie Get Married Facebook page.
A statement from the show's makers said: "They remain puppets and do not have a sexual orientation."
But they conceded that the pair are "male characters and possess many human traits and characteristics".
The confirmed bachelors have lived together for 40 years and sleep in the same bedroom, albeit in single beds.
