Retired film lighting technician Thomas Markle, who currently lives in Mexico, is said to be "desperate to heal the rift" - before his 37 year old daughter gives birth.
In an exclusive story published by The Mirror newspaper, a source said: "He [Thomas] strongly believes the baby should be as much a part of his life as they will be Prince Charles's."
The article claims the 74 year old has appealed for help from the Archbishop of Canterbury - who married Prince Harry and Meghan during a service at St George's Chapel, Windsor in May 2018.
It alleges that he has also sought help from Kensington Palace - the official residence of the Duke and Duchess of Sussex.
Sources referred to by The Mirror claim Mr Thomas has hired lawyers to secure permission from the courts to ensure he has contact with his future grandchild.
Prince Harry and Meghan announced last week during their tour of Australia, New Zealand, Fiji and Tonga that they were expecting a child.
It is understood the duchess has not spoken to her father since her May wedding, which Thomas was unable to attend. He was declared too unwell to fly after suffering heart palpitations.
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