Prince William is said to be hoping the Duke and Duchess of Sussex "are all right" after they told an ITV documentary they were struggling.
The palace source said there was a view the couple were "in a fragile place".
Kensington Palace had no comment on the ITV film, which aired on Sunday.
The documentary followed the Duke and Duchess of Sussex on their tour of southern Africa earlier this month.
In interviews, the couple both said they were struggling with the intense scrutiny from elements of the British tabloid press.
The palace source played down suggestions that Prince William was "furious" with his brother after the documentary aired on Sunday.
It followed Prince Harry's admission in the film that he and William have "good days" and "bad days".
Asked about media reports of a rift between the brothers, the Duke of Sussex said that "inevitably, stuff happens" as a result of "this family being under the pressure that it's under".