Describing the period as one of the most painful of her life, the 67-year-old singer made the admission during her first podcast interview, a two-hour conversation with On Purpose with Jay Shetty.
She also spoke about how spirituality helped her endure dark moments, including the legal fight with her former husband Guy Ritchie in 2016.
Madonna said: "There were moments in my life I wanted to cut my arms off… I actually contemplated suicide.
"I would say probably one of the most painful moments in my life where I honestly couldn't see the forest for the trees was when I went through a custody battle with my son.
"Even though my marriage didn't work out… I mean a lot of people's marriages don't work out. They marry the wrong people. They're not aligned. They're not meant for each other. Someone trying to take my child away from me was like, they might as well just kill me. That's really how I was thinking.
"And I was on tour at the time, so I had to go on stage every night. I would just be lying on the floor of my dressing room sobbing. I really thought it was like it was the end of the world. I couldn't take it. I just couldn't take it."
Madonna became involved in the custody dispute when Rocco, then 16, chose to remain in London with his father while she was touring with Rebel Heart.
After a lengthy legal process, a settlement was reached that allowed him to stay in the UK.
She and Guy had divorced in 2008.
The singer, who has six children, said she is now "good friends" with Rocco.
She credited her faith for helping her endure:
Madonna added: "But thank God I don't feel that way anymore … I'm happy to say that I'm really good friends with my son but I couldn't see it then. I really thought it was the end of the world. So, you know, thank God I had a spiritual life."
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