The presenter is in addition to current presenters Monty Don, Carol Klein, Joe Swift and Rachel de Thame.
The show has been running since 1968. Don has been lead presenter since 2011.
The BBC has yet to announce its 2016 schedule, saying that will be revealed in the second or third week in January.
Klein said last year that the BBC did not give presenters word that they would be working on the show, which usually starts in late February and runs until November, until the New Year.
The advert states: "BBC Bristol are currently looking for potential presenters for a new series of Gardeners’ World. Previous television experience is desirable but not essential, as long as you are an enthusiastic and passionate communicator about plants and gardening. Contact Helen Cawley, email:firstname.lastname@example.org or tel: 0117 974 2309 or 07729 260633 before January 22nd."
The position is "additional" to the current presenters, who the BBC see as a "good solid team".
A BBC spokesperson said: "The note has just been put out to see who’s out there, that’s it."