Rimes ended her marriage to Dean Sheremet several years ago after meeting Cibrian, who was at the time married to Brandi Glanville.
The country music star has opened up to E! News about the guilt she feels over how she handled her split from Sheremet.
"I don't like the word regret. I wish I handled it differently because I wish I could have been better for me, for Brandi, for Dean, for Eddie, for everyone else," she said.
Rimes continued: "I know, in this situation, it's going to take some time. All I wish is that everyone that was hurt, that we hurt, that I hurt, can be happy… No-one will ever understand how much thought and hurt were put into our decision."
Tension between Glanville and Rimes has remained, with the Real Housewives of Beverly Hills star recently publicly complaining about Rimes's treatment of her and Cibrian's sons.
Glanville has since vowed to no longer air her grievances with Rimes and Cibrian publicly,