The versatile player began his career in the Exeter City Academy, joining at the age of eight and progressing through the ranks before signing his first professional deal in April 2014.
During the 2014-2015 season, he made one appearance for the Grecians as a substitute in an EFL Trophy while being an unused substitute in League Two four times.
In July 2015, he made a loan move to the Seagulls, making 25 appearances and being an unused substitute six times, with one goal and two assists coming in the 2015-2016 campaign.
After being released by Exeter in the close-season of 2016, Pope rejoined the Seagulls on a permanent deal just ahead of the 2016-2017 season. During his three-year stay in North Somerset, Pope made 138 appearances while being an unused substitute seven times, getting one goal and assisting ten.
The versatile player moved onto National League North side Hereford FC in June 2019, where he made 34 appearances and was an unused substitute seven times, scoring five goals and assisting two during his time at Edgar Street.
In August 2020, Jason signed for National League South club Bath City. During his time at Twerton Park, Pope made 11 appearances and was an unused substitute eight times before departing after the season was declared null and void.
Pope returned to the Seagulls for a third spell in May 2021 ahead of the 2021-2022 season and forged a strong partnership with Sam Avery during 2021-2022. Jason was part of the Seagulls side that saw the club win the Southern League Premier South title in 2023 and return to the National League South after four years away.