In a statement Thursday, they asked for "consideration and respect for our family as we go through this difficult period."
The 35-year-old actress gave birth to twins - a son, Miller, and a daughter, Johnnie Rose - in 2006. The couple used an anonymous donor from a sperm bank.
The 48-year-old rocker also shares custody of two other children with former partner Julie Cypher. Cypher. Musician David Crosby was the sperm donor for the two children.
Melissa Etheridge was diagnosed with breast cancer in 2004.
The Associated Press contributed to this report.