Proposals for the upgraded tennis court at Anmer House would see it moved from its current position to improve views from the hall.
The new court, on the Queen's Sandringham Estate in Norfolk, would would be resurfaced with astroturf, says the application.
King's Lynn Borough Council will consider the plans at a later date.
The application, lodged by the Estate Office at Sandringham, says the "current tennis court needs extensive work to resurface" and it would make sense to "remove all the existing court and start again".
It says: "On completion of the new court, the garden will be prepared to reinstate the fencing and re-erect parkland iron fencing."
The Royal couple are known to be fans of tennis and have previously been spotted attending games at Wimbledon.
Plans to revamp the tennis court at the Duke and Duchess of Cambridge's country mansion have been submitted.