They could be away for two or more years.
A move overseas would follow a year in which Harry and Meghan have hardly been off some front pages.
The idea to move Harry to Canberra was dismissed as impractical, The Sunday Times reports, but courtiers are now reportedly drawing up plans to hand Harry and Meghan a major worldwide job in Africa after the birth of their child.
In a statement previously released by Kensington Palace in October, Doria, 62, shared her reaction to her daughter's pregnancy announcement as the rest of the world learned Meghan, 37, and Harry, 34, are expecting their first child together.
The Duke and Duchess used their new Instagram account to post a series of throwback snaps of the monarch - and some showing Harry in his younger days. "Harry & Meghan", they wrote.
Zookeeper attacked by tiger, transported to hospital in Kansas
Elephant staff members were a short distance away and were able to lure the tiger back into its enclosure, where he was secured. The director credited his staff with a speedy and effective response that got the incident under control within 10 minutes.
The cost of the SO14 unit who protect the Royal Family has been estimated at around £130million, although experts say looking after the Sussexes in Africa could cost an extra £1million.
The Sussexes are thought to have fallen in love while on a safari in Botswana, returning there for Meghan's 36th birthday two years ago. South Africa and Malawi are other options, according to the newspaper report.
The Palace has not denied the couple could be moving overseas.
A spokesman for Harry and Meghan said: "Any future plans for the duke and duchess are speculative at this stage".
The talk follows almost a year of rumblings of a feud between the royal couples which has only been spurred on by the Sussex's recent move to Windsor and an earlier choice to break away and separate their royal household from the Cambridges'.
On Sunday, Buckingham Palace revealed no decisions about Meghan and Harry's future roles have been made.