The 25-year-old Leathersich had Tommy John ligament replacement surgery in July. He was 0-1 with a 2.31 ERA with 17 relief appearances for New York last season, his first in the majors.
The Cubs also agreed to a contract for next season with right-hander Andury Acevedo on Thursday. He spent the past four years in the New York Yankees organization and reached Triple-A Scranton/Wilkes-Barre last season.
The moves put the Cubs' 40-man roster at 32.