The 24-year-old, who has been nominated for an Academy Award for her performance in An Education, is said to be in negotiations with the producers of the upcoming musical after actress Keira Knightley turned down the role last month.
Daily Mail columnist Baz Bamigboye said: "The actress has not been formally offered the part yet - and won't be until Columbia Pictures agrees the film's budget.
"Carey and the My Fair Lady filmmakers will keep talking until a deal can be struck."
According to reports, Mulligan has been preparing for the role by taking singing lessons to help perfect her portrayal of the character. While, movie bosses are rumoured to be eyeing actor Colin Firth to portray Professor Higgins.