The pair, who have been dating since 2007, allegedly spent the past few weeks trying to save their relationship, but realised that they are living separate lives due to their busy schedules.
"Things haven't been going well and they haven't seemed like a couple for the past few weeks. At Glastonbury festival, they barely said a word to each other," a source told the Sunday Mirror.
"They really wanted to give it a go to hold it together and seemed to be getting back on track - but things got bad again and they both decided enough was enough. They want to stay friends so it's best to split now.
"They've been through a lot together and all their friends are sad they've called time as a couple. Alexa and Alex both wanted things to work. It was best that they both got out now and saved their friendship."
Chung's spokeswoman Liz Matthews confirmed last night: "It's completely amicable - but yes it's true. They have split."
Turner moved to New York with Chung while she hosted her own MTV show, but is now expected to move out of her apartment.
> Alexa Chung to host a fashion TV show