Lawyers for the the royal husband and wife claim the letters publication was a misuse of private information and breached her copyright.
The publication of the letter is claimed to have caused a deep rift between Meghan and her 75-year-old father.
Meghan and Harry are seeking aggravated damages from the paper.
Documents from the Duchess' lawyers this week accused the Mail and other tabloids of harassing, humiliating and manipulating the 38-year-old former Suits star.
The lawyer for the newspaper, Antony White, is pushing to have parts of Meghan’s claim dismissed, saying they were irrelevant or impermissible, not properly pleaded or disproportionate for the court to investigate.
A source claims that Meghan and Harry - who have moved to Los Angeles after stepping down as senior members of the royal family on March 31 - are expected to listen in remotely.
The hearing is one of the first stages in the legal action and a date for a full trial has not yet been set.