A source said Markle is "quite opinionated and Harry has got very dictatorial of late, which hasn't made things easy at times".
Prince Harry and Meghan are moving to Windsor early next year.
The Duke and Duchess's official office will continue to be based at Kensington Palace.
Before Harry and Meghan can move into the 10-bedroom cottage, it will need substantial renovations since it's now used housing for staff.
Grade II listed Frogmore is said to be undergoing a multi-million pound refit, paid for by the taxpayer, in preparation for the birth, which is expected next spring. However, the British media reported that the move caused tensions between Harry and his brother Prince William, who also lives with the family at Kensington Palace. They now live in two-bedroom Nottingham Cottage at Kensington Palace.
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The parents-to-be are funding the structural and landscaping elements of the project with money from the Sovereign Grant, according to Kensington Palace, while they'll be footing the bill for the decor themselves.
Prior to this report, a royal source revealed that Prince Harry and Prince William had some tension.
Windsor is a "very special place" for the couple for obvious reasons but the move has led to speculative reports of a rift between Harry and William, as well as their wives.