Galliano was forced out of the job he had held for almost 15 years after he was accused of launching into a racist and anti-Semitic tirade at a Paris bar in March.
Jacobs already heads up the Louis Vuitton brand as well as his own Marc Jacobs line of clothing.
Dior is owned by French luxury group LVMH, which already owns Louis Vuitton where Jacobs has been creative director since 1997, helping lead the label to huge global success.
WWD said that Phoebe Philo, currently head designer at Celine, would be ready to step in to the top Louis Vuitton job if Jacobs makes the move.
Speculation has been rife about the top job at Dior since Galliano was dismissed in March. Names in the frame have included Christian Lacroix, Alexander Wang and the designer of Kate Middleton's well-received wedding dress, Sarah Burton.
John Galliano appeared in a Paris court in June to face charges of "public insults based on origin, religious affiliation, race or ethnicity." Galliano blamed the incident on the cocktail of sleeping pills, alcohol and valium which he had been taking to cope with the demands of his work. A judgment is expected on September 8th.