Josh, now 27, has been accused of molesting five underage girls in 2002, when he was just 14. Within hours, the "19 Kids and Counting" star admitted, in a Facebook post, to acting "inexcusably".
The interview would air during host Megyn Kelly's "The Kelly File" on Wednesday, June 3.
Additional portions will air during a one-hour special on Friday, June 5.
The duo released a statement on Michelle's blog.
"Next week we will sit down with Megyn Kelly on Fox News to share our hearts with you about the pain that we walked through as a family twelve years ago, the tears we all shed and the forgiveness that was given. We appreciate the outpouring of love and prayers for our family at this time."
"19 Kids", an unscripted program, which follows parents Jim Bob and Michelle and their 19 children - all of whose names begin with the letter J - first launched in 2008. It is TLC's most successful program.