The pair are said to have met in May while Harry, 32, was in Canada promoting Invictus Games Toronto 2017, the paralympics-style competition for injured servicemen and women which was started by the Prince.
Markle, 35, star of hit US TV show Suits, is said to have visited the UK several times in recent months.
According to the Daily Express, the pair share a passion for Africa, where Harry recently spent time in Malawi, and where Markle has carried out humanitarian work.
Pictures have also emerged of the pair appearing to wear identical bracelets.
Kensington Palace has declined to comment and Markle, a regular user of social media, has not confirmed or denied the reports.