TR Sullivan of MLB.com reported Jones underwent surgery Wednesday and should return for next year's spring training.
Jones won the 2019 True Ranger Award as the minor league player who best represented the "core values of the organization in a positive light both on and off the field."
The left-handed reliever posted a 2.67 ERA and 1.16 WHIP with 71 strikeouts in 64 innings across the Double-A and Triple-A levels last year.
He was competing for a spot in the Rangers' bullpen for 2020 during spring training before the season was delayed indefinitely because of the coronavirus pandemic.
The 31-year-old New York native had allowed just one run through 4.2 innings (1.93 ERA) across four spring outings.
Jones will become a free agent following the 2020 campaign after signing a minor league contract in January.