The 60-year-old Jauron spent the past two seasons coaching Philadelphia's secondary. He previously coached in Buffalo and Chicago, posting a 60-82 record.
Jauron was a candidate for Philadelphia's defensive coordinator position after the team fired former coordinator Sean McDermott, and the Eagles hoped to retain Jauron, according to the source.
Shurmur, hired last week as Cleveland's fifth coach since 1999, is also looking for an offensive coordinator after Brian Daboll was fired and signed with Miami. Rob Ryan, Cleveland's defensive coordinator the past two years under Eric Mangini, is now with the Dallas Cowboys.
The Browns also are in the market for an offensive coordinator. Shurmur plans to call his own plays next season, which could be limiting his choices.
The Associated Press contributed to this report.