Speculation about the pair's relationship began earlier this week when Kinnaman took Munn to the second season premiere of his AMC show The Killing.
E! News has now reported that the couple began dating shortly before Christmas.
A source claimed that the pair "just get each other" and revealed that they are "extremely happy together".
"They are on the same wavelength intellectually, and they have the same sense of humour," the insider said. "It's been going on for a little while and it's heading towards being serious."
As well as starring in The Killing, Kinnaman will appear in a new remake of RoboCop. Meanwhile, Munn is currently working on HBO's new Aaron Sorkin drama The Newsroom.