The 32-year-old former tennis player is said to be growing increasingly frustrated with the 'Hero' singer's reluctance to marry her and is ready to ''move on'' after being in a relationship with the musician for 12 years.
A source told the New York Post newspaper: ''[They're] on the rocks. Anna is getting tired of waiting for him to decide if he wants to marry her or not. They've been together for so long, she feels like it's never going to happen. She is ready to move on.''
However, the insider is hopeful Enrique, 38, and Anna, can work through their problems as it isn't the first time they have struggled with the issue.
The source added: ''This issue has come up before, and they have managed to smooth things over and get back together. Time will tell.''
Enrique - whose father is legendary crooner Julio Iglesias - has previously spoken of his belief that marriage isn't important.
He has said: ''I've never really thought marriage would make a difference. Maybe it's because I come from divorced parents, but I don't think you love someone more because of a piece of paper. And nowadays, it's not taboo to have kids and not be married. What makes a difference is that you're a good parent, period.''
Ahead of his new album 'Triplicate', Bob Dylan reveals his favourite artists.
'Mindhorn' sees Julian Barratt as a former TV star who pretends to be a detective to nab a killer.
Iron Fist co-creator Roy Thomas 'tries not think' about the critics of the Netflix/Marvel series, because he has 'so little patience' for them.