The outlet is reporting that the titles that they now have can be passed on to any boy children they have - meaning that their sons will be princes or dukes - but their girl children will not be princesses or duchesses.
According to the Palace, the two women will spend a busy day together next week in the English city of Chester.
The Palace has confirmed Meghan will join The Queen for a day of royal engagements next week.
It will come as a serious blow for former Suits actress Meghan, who has long championed women's rights and recently declared in a biography published on the monarchy's official website: "I am proud to be a woman and a feminist".
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The royal laws of succession were changed to become gender-neutral in 2013 - meaning the Duke and Duchess of Cambridge's daughter Princess Charlotte became the first female royal to retain her place in the line of succession.
Group shot: The family snap with Meghan centre stage.
The Duchess of Sussex Meghan Markle will accompany the Queen next week on a visit Chester where her majesty will formally open the Storyhouse.
The source added that Prince Harry told his new mother-in-law "numerous stories about the Queen as a grandmother and what she (means) to him".
However, during his down time with Meghan, Prince Harry made an exciting reveal about the Invictus Games.