updated 03/13/2014 AT 6:00 PM ET
•originally published 03/14/2014 AT 1:55 PM ET
“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 in January.
“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.”