The head of state announced there would be a "period of transition" with Harry and Meghan - who have said they want to be financially independent and step back as senior royals - living in Canada and the UK.
She stressed the discussions were "complex matters" for her family to resolve but she wanted "final decisions" in the "coming days".
The full statement read: "Today my family had very constructive discussions on the future of my grandson and his family.
"My family and I are entirely supportive of Harry and Meghan's desire to create a new life as a young family. 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.
"Harry and Meghan have made clear that they do not want to be reliant on public funds in their new lives.
"It has therefore been agreed that there will be a period of transition in which the Sussexes will spend time in Canada and the UK.
"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."