According to paper, some are unhappy about the fact that the Duke and Duchess of Cambridge are not club members.
Founded in 1869, rules state that non-members may only play on the court if directly invited by another member - something that William and Kate appear to have bypassed.
The source added: "Unfortunately some patrons feel Charlotte shouldn't be allowed to play there.
"Club rules are usually very rigid, and they don't see why exceptions should be made, even if they happen to be royal."
MailOnline has contacted Kensington Palace for a comment.
Charlotte's interest in the sport comes as no surprise with her mother being a keen tennis fan and patron of the London Tennis Association, having taken over the post from the Queen.
The Queen was patron of the LTA, which is the national governing body of tennis in Great Britain, the Channel Islands and the Isle of Man, for 64 years.