Prince Harry, Prince William, the Duke of Cambridge and their father Charles, the Prince of Wales, will all attend the meeting, while Meghan was expected to join the discussion by phone from Canada, reports the BBC.
It is hoped the talks will produce a "next step" on the way to defining the couple's new relationship with the Royal Family - in line with the Queen's wish to find a solution within days.
Among the issues likely to be discussed are what funding the Duke and Duchess of Sussex will receive, whether they will keep their titles, and what royal duties they will continue to carry out.
Monday's gathering at the Queen's estate in Norfolk - being described as the "Sandringham summit" - will be the first time the monarch has come face-to-face with Prince Harry since the Sussexes' announcement, which was posted on their official Instagram account on January 8.
But newspaper reports suggested the couple could conduct a potentially damaging television interview if they were unhappy with the outcome of the discussions.
Broadcaster and friend of the couple, Tom Bradby, who made a documentary with them last year, wrote in the Sunday Times: "I have some idea of what might be aired in a full, no-holds-barred, sit-down interview and I don't think it would be pretty."
Meghan is currently in Canada with her eight-month-old son Archie after flying there amid the ongoing discussions, which have involved the UK and Canadian governments, reports the BBC.
The couple had been in Canada over Christmas, before they returned to the UK last week after a six-week break from royal duties.
Last October, Harry and Meghan publicly revealed their struggles under the media spotlight.
The couple were already preparing to launch their own Sussex Royal charity, which they set up after splitting from the Duke and Duchess of Cambridge's foundation in June last year.
In December it was revealed that the couple had made an application to trademark their Sussex Royal brand across a string of items including books, calendars, clothing, charitable fundraising, education and social care services.
( With inputs from IANS )