"There can be nothing more impressive for a young boy than seeing that," says the source.
Up until now, Harry, along with his wife Meghan, Duchess of Sussex, have been hell-bent on telling "their truth" to US media outlets.
But insiders say Harry is now realising their vendetta may result in Archie and his sister Princess Lilibet missing out on a relationship with their paternal grandfather - especially given the Sussexes remain estranged from Meghan's dad, Thomas Markle Sr.
New Idea now understands Harry, 38, is trying to convince Meghan, 41, they should at least mend fences with Charles, 74, who shares a warm relationship with his Wales grandchildren, Prince George, Princess Charlotte and Prince Louis.
WATCH: Prince Harry arrives at High Court in London
"Aside from [Meghan's mum] Doria, Charles is the only grandparent Archie's got," says the source. "Harry has noticed Charles is softening with age, and has become a perfect grandpa to Wills' kids. Harry wants that for Archie and Lili, but knows it would mean more frequent visits to the UK - something Meghan's not keen on."
With the King's first Trooping the Colour set for June 17, a separate insider reveals Harry is eager to have Archie experience the exciting day. Given Lili is only 1, he feels she is still too young.
"Meghan will put her foot down, she has many meetings and projects on the go, but Harry won't let Archie miss out on another opportunity," says the source, adding Harry is determined to make it happen and put the past behind him once and for all.
For more, pick up the latest issue of New Idea!