The pair are said to have been spotted on a series of dates, and are planning a holiday in Mexico together, reports The Sun.
They were allegedly seen on a dinner date at Hotel Bel-Air's Wolfgang Puck restaurant in Los Angeles last week.
A source said: "They are very comfortable together and very into each other. Bradley couldn't keep his eyes off her.
"She knows how to work her charm."
The couple were previously romantically linked back in September when they were spotted together.
Scarlett Johansson recently stated that her divorce from actor Ryan Reynolds was "horrible".
Bradley Cooper has signed up to star in The Hangover Part III, which will return to the setting of Las Vegas.
Watch Scarlett Johansson in an Avengers featurette below: