Jonathan Greenstein was born in Tel Aviv. He discovered jazz at fourteen, when an eighth-grade music teacher gave him a cassette of Charlie Parker's Bird with Strings. He studied at Berklee College of Music (2007–2009), earned a master's degree at the City College of New York (2013–2015), where he worked as a teaching assistant under saxophonists Steve Wilson and Scott Reeves, and completed a Doctor of Musical Arts at the University of Wisconsin–Madison (2017–2020), with a dissertation titled Astral Fire: A Meditation on Spiritual Jazz as Modeled by Five Saxophone Players, 1969–1979, independently catalogued by ProQuest.

His debut album, Thinking, was released in 2011 on the respected Fresh Sound New Talent label (catalog FSNT-386), recorded in October 2010 at Wellspring Studios in Acton, Massachusetts, with guitarist Ilan Bar-Lavi, pianist Victor Gould, bassist Dan Carpel, drummer Jeff Fajardo, and a guest appearance by trumpeter Darren Barrett, winner of the Thelonious Monk International Jazz Competition. Haaretz critic Ben Shalev reviewed the album favorably in July 2011, and it later received a substantial independent feature in PopMatters in 2018. He went on to self-release a series of EPs on his own Rabbit Rabbit Rabbit Records — Vol I (2016, featuring guitarist Gilad Hekselman), Vol II (2017), Vol III (2019) — alongside a standing residency band at Rockwood Music Hall in New York.

He has held teaching posts at City College of New York, the Wisconsin Conservatory of Music in Milwaukee (from 2018), and, in Israel, as a guest instructor at the Israeli Conservatory of Music's Center for Jazz Studies in Tel Aviv (2015–2018) and at Thelma Yellin High School for the Arts (2016). His own résumé lists performance credits alongside bassist Omer Avital and pianist Shai Maestro in New York, and a 2012 appearance at the Tel Aviv Jazz Festival alongside baritone saxophonist Ronnie Cuber. He continues to record as a sideman, most recently on Daniel Meron's 2024 release Pendulum (Pinch Records).

Greenstein's academic focus — a doctoral dissertation on spiritual jazz saxophonists of the 1969–1979 period — sits close to his own playing, which draws on the post-bop and spiritual-jazz lineage that dissertation examines. His debut composition on Thinking was co-written with trumpeter Itamar Borochov, and the record's guest slot for Darren Barrett situates him within a network of Berklee-connected players from the same generation.