She said she was "entirely supportive" of their desire for a new role but "would have preferred" them to remain full-time working royals.
She expected final decisions to be made in the coming days, she said.
Senior royals have been in talks about Prince Harry and Meghan's role after they said they wanted to "step back".
In a statement, the Queen said the talks at Sandringham, which also involved the Prince of Wales and the Duke of Cambridge, had been "very constructive".
"My family and I are entirely supportive of Harry and Meghan's desire to create a new life as a young family," she said.
"Although we would have preferred them to remain full-time working members of the Royal Family, we respect and understand their wish to live a more independent life as a family while remaining a valued part of my family."
She said it had been agreed there would be "a period of transition in which the Sussexes will spend time in Canada and the UK" after Harry and Meghan "made clear that they do not want to be reliant on public funds in their new lives".
"These are complex matters for my family to resolve, and there is some more work to be done, but I have asked for final decisions to be reached in the coming days," she said.
The urgent talks were convened after the Sussexes surprised the rest of the Royal Family on Wednesday with a statement expressing their desire to "step back as 'senior' members of the Royal Family".
They also said they wanted a "progressive new role" within the institution, where they would be financially independent and divide their time between the UK and North America.
Although no other family member was consulted about the timing of the announcement, the duke and duchess said it came after "many months of reflection and internal discussions".
Both Prince Harry and Meghan spoke of the difficulties of royal life and media attention in recent months, with the duke saying he feared his wife would fall victim to "the same powerful forces" that led to his mother's death.
The talks about their future took place as Prince Harry and Prince William issued a joint statement denying "false claims" that their relationship had been damaged by "bullying" on the part of the older brother.
They said that the "inflammatory language" in the claims was "offensive" and "potentially harmful", given their support for mental health causes.