Elizabeth Vargas and Marc Cohn
Matt Baron / Beimages
03/14/2014 AT 01:55 PM EDT
"Walking in Memphis" singer Marc Cohn is responding to a report he cheated on his wife Elizabeth Vargas
while she was in rehab
"I will not comment on what is an extremely sensitive and private family matter – addiction and recovery are serious challenges and it's been a tough road for all of us," Cohn tells PEOPLE in a statement. "I have been supportive of Elizabeth in the past – and will continue to be – regardless of the long-standing issues in our relationship."
The New York Post
reports the Grammy-winning songwriter had an affair while the 20/20
anchor was seeking treatment for alcohol abuse over the holidays.
A rep for Vargas had no comment.
A source tells PEOPLE that Cohn "has a strong emotional investment in how the story is told because his primary concern are his kids and he doesn't want to look like a bad guy when he's not." The source adds: "It's a complicated situation since what seems new in the public is a longstanding challenge for them over the years."
Cohn, 54, and Vargas, 51, have been married since 2002 and have two sons
, Zachary, 11, and Samuel, 7½.
Vargas announced that she was seeking treatment in November, and returned to TV
"I am an alcoholic," she said in an interview on Good Morning America
. "It took me a long time to admit that to myself. It took me a long time to admit that to my family. But I am."