The 32-year-old Captain America actor is said to have had a crush on Sandra Bullock since he was in middle school. Sources tell Us Weekly the two are now friends, and that Evans is actively "pursuing" the 49-year-old actress.
"[Chris is] totally her type!" a different source adds.
Bullock was married to Jesse James for nearly five years, but the couple divorced in 2010 amid rumors that James had been unfaithful. The actress has not publicly dated since, and Evans has been linked to Minka Kelly in the past.
The magazine reports Bullock has not let the relationship "go to the next level yet." The pair frequently go out together in Los Angeles, and "spent the night talking until 12 a.m." at a dinner with friends earlier this year.
Evans will next appear in Avengers: Age of Ultron, while Bullock is set to star The Heat 2.