The 44-year-old designer has led both the creative and business sides of the British fashion house since the departure of former chief executive Angela Ahrendts in 2013, but executives are keen to bring in some support to assist with the workings of the company.
A shareholder told the Financial Times: "Bailey needs back-up. He needs someone to help him on the marketing and retail side, who has a good understanding of the business and knows exactly where they want to take the company."
There is said to be no question over Christopher's experience and skills as creative director, but investors are keen for him to focus on his area of expertise by bringing in someone else to complement his work by taking the reins of the business side.
Another shareholder said: "It worked with Christopher Bailey as head of design and Angela Ahrendts as chief executive, but there have been problems since he took up both roles after she left."
No names up for the rumoured position have been disclosed as yet, while the fashion house - whose shares have fallen 21% since May 2014 and are down 32% over the last 12 months - haven't yet commented on the reports.
Bailey joined Burberry 15 years ago as design director and was appointed creative director three years later in 2004, and chief creative officer in November 2009.
In October 2013, it was announced he would be the next chief executive and took up the position on May 2014 when Ahrendts departed for Apple.
