Lombardi was named the Browns' vice president-player personnel in January.
Tom Heckert was fired as general manager Dec. 31, the same day Cleveland fired Coach Pat Shurmur. Rob Chudzinski was hired as coach Jan. 10.
Lombardi, no relation to NFL coaching legend Vince Lombardi, has been an executive for five other NFL teams before joining the Browns. Most recently he was senior personnel executive from 1998 to 2007 for the Oakland Raiders.
The Browns said Tuesday Farmer's duties include helping Lombardi manage the professional and college scouting departments. Farmer was the director of pro personnel for the Kansas City Chiefs (2006-12).