Moore, 50, is the estranged wife of actor-producer Ashton Kutcher.
"They had a lot of fun together and got on really well," a source told E! of Moore and Morton, 31.
"And realized in the getting to know one another process that they have a lot of mutual interests. But after dating for six months or so they both realized that while they had become very good friends through the dating process, this wasn't going to turn into a long-term romance. ... They quietly stopped having dates together, but they have remained in regular contact as they get on so well. ... You will occasionally see them together as friends, but they have stopped seeing each other romantically."