A spokeswoman for Meghan had no immediate comment on the report.
According to the Times, Meghan recorded the voiceover before the royal couple left for their six-week break to Canada to celebrate Christmas.
In Washington, President Donald Trump weighed in, telling Fox News that he found the spat "sad". "But I find it, I just have such respect for the Queen". "She said, "hey, I want to be a princess" so now she's a princess", she added.
"We understand their desire to take a different approach, but these are complicated issues that will take time to work through", that statement noted.
The source said officials had consulted the British and Canadian governments about the issue of the couple's future.
The process was expected to take "days, not weeks", the source said. Their statement was a shock to many including to Queen Elizabeth II and Prince Charles as the couple didn't inform them of their plans ahead. Prince Harry and Markle have reportedly told the Prince of Wales that they wanted to live largely in Canada but wanted to keep their home in Frogmore Cottage. The brothers were close for years following the death of their mother Princess Diana in a Paris vehicle crash in 1997.
With their wives, Kate and Meghan, they have been viewed as the modern face of the royal family, hailed for bringing fresh energy to the institution.
Meanwhile, Prince William has expressed his "sadness" over the rift with his brother Prince Harry and that the royal family was no longer "a team", the UK's Sunday Times reported, citing a friend.
The couple had returned to the United Kingdom earlier this week to meet Canada's high commissioner, to thank them for their six-week break in the country over Christmas.
The decision by the couple, who recently registered the trademark Sussex Royal, to effectively resign their royal roles follows a turbulent year for the Windsors.
In an emotional TV interview last October, Harry admitted he and William had fallen out. Trump felt it wasn't fair to Queen Elizabeth, who he considers as a great woman.
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