"He's happier than he's been for many years," a source told The Sunday Express.
Although the source says the couple is taking the relationship slowly, Harry is said to be "besotted" by the brunette beauty.
"They are taking each week as it comes and just enjoying each other's company but it's fair to say that they love seeing each other and there's a definite chemistry between them," the insider said.
"Harry has been desperate to keep the relationship quiet because he doesn't want to scare Meghan off. He knows things will change when their romance is public knowledge but he also knows he can't keep it a secret for long."
More From This Section
The 35-year-old actress has posted several pictures on Instagram from her trips to London over the summer.
"It's too early to say if the relationship will lead to anything long-term but who knows" the source said. "At the moment they are just taking it a step at a time and seeing how things develop."
Markle's rep did not respond to the request for comment, and a Kensington Palace spokesperson says they "would not comment on private matters.