According to reports, the 26-year-old actress is "smitten" with the 50-year-old actor after getting together in January while she was staying at the Chateau Marmont hotel in Los Angeles.
A source told Us Weekly: "Their first hook-up was in Scarlett's bungalow. She is smitten with him."
Johansson and Penn allegedly began their friendship when she was planning to fly out to Haiti to help those affected by the two earthquakes in January 2010.
The insider added: "Scarlett first reached out to Sean when she was planning to visit Haiti with Oxfam. She knew Sean was living there in a tent and turned to him for advice."
Johansson did not make the trip, but the pair have apparently developed feelings for one another.