The "Avengers: Age of Ultron" co-stars have been dating for months, reported Ace Showbiz.
"Chris and Lizzie are definitely involved in some kind of ultra-secret romance that's being kept under wraps while they shoot," a source said.
"The attraction between them has been there from day one."
The 34-year-old "Captain America: The First Avenger" actor allegedly wants his personal life to be very private.
"Chris is paranoid about his womanising becoming public while he's on this multi-picture deal with Marvel. He's protecting his image, and will bend over backwards to keep things quiet," the source added.
Olsen, 26, ended her engagement to Boyd Holbrook in January after two years together.
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