The pair are believed to have been on a series of dates together after growing closer earlier this year.
Their alleged secret dates are said to have included a romantic night out in New York last weekend.
A source told The Sun: "They've had a flirtation for years and have always fancied each other, but since May this year they've been talking a lot.
"Finally their schedules allowed them to spend a weekend in the same part of the world and they made the most of it.
"They are both really busy but Rihanna has asked him to come to Rock in Rio in September so he can see her perform and spend time together."
And a source close to Lewis added: "He is a bit infatuated with her and thinks she is not only sexy but incredibly funny, although he knows she has a reputation as a heartbreaker."
Rihanna's most significant relationship has been with Chris Brown which was notoriously troubled.
Lewis' seven-year relationship with singer Nicole Scherzinger ended in February.
While Lewis was recently linked to Keeping Up With The Kardashians star Kendall Jenner, Rihanna was believed to be enjoying a flirtation with Leonardo DiCaprio...
RiRi is currently working on her eighth studio album.
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