The new apartment is said to be much closer to the huge wing that will be the permanent home for the Duke and Duchess of Cambridge from September, when Prince George will start at Thomas's school in Battersea, South London. "It's sweet that Harry and Meghan will be living next to William and Kate," adds the source.
Kensington Palace spokesmen have refused to comment on suggestions that an engagement announcement for Harry could be imminent.
Meghan, who divorced her first husband after a two-year marriage, is said to be keen to have children with Harry. After their romance was disclosed last spring, the Prince took the rare step of confirming Meghan as his girlfriend in a statement.
He had never spoken publicly about his courtship of aspiring actress Cressida Bonas or his on-off romance with law graduate Chelsy Davy.
William and Kate have 22 rooms and two kitchens in their lavishly refurbished apartment. A palace spokesman declines to comment on Prince Harry's new apartment, describing it as a "private matter".