Prince Charles is understood to have made a similar arrangement with one of his former bodyguard's to protect the Duchess of Cornwall when she was still Camilla Parker Bowles.
Meghan is said to think Harry's idea is "charming", but unnecessary.
It is understood the cost of such protection would be around £50,000 a year.
In a strongly worded statement last week, Harry accused the media of intrusion and lashed out at the "sexism and racism of social media trolls".
His message, via his communications secretary Jason Knauf, was that their relationship "is not a game - it is her life and his".
The statement added that Harry was "worried about Miss Markle's safety" and "deeply disappointed that he has not been able to protect her."
It was reported yesterday that divorcee Miss Markle has fallen "head over heels" for the prince, with a friend saying she will "give up all that she is doing" in order to make their relationship work.
The pair have made a number of trips to see each other although they have not yet been seen together in public.
Prince Harry jetted off to Canada to stay with Miss Markle last month and the actress is believed to have returned the visit with a mini-break in London just last week.
She was seen shopping in upmarket Whole Foods on Kensington High Street, just down the road from Kensington Palace.
The visit is the clearest indication yet that the relationship between the divorcee and the fifth-in-line to the throne is serious.
