They are adapting Fey's 2004 film which starred Lindsay Lohan, Rachel McAdams, Amanda Seyfried and Lacey Chabert as a cliquey gang of mean high-school girls.
The film also starred Lizzy Caplan, Amy Poehler and Fey herself as a teacher.
Richmond told Billboard: "We definitely want to see some kind of shape by the end of the summer."
But he said there would still be plenty of work to do once he, Fey and lyricist Nell Benjamin felt they had finished.
"We know that 60 or 70% of that will go away when we start bringing in real singers and actors. We'll do a lot of rewriting," he said.
He added: "The most difficult part of the process is determining what are your best song ideas, and what do you leave alone as dialogue?"
The musical is eagerly-awaited by fans of the film, which won several MTV Movie Awards and Teen Choice Awards in 2005 and has since gained cult status among many.
Some of the film's best lines - such as "fetch" and "you can't sit with us" - have also been adopted into many fans' everyday vocabulary.
The Mean Girls musical by Tina Fey and her husband, composer Jeff Richmond, should be finished this summer, according to Richmond.