Kent finished second in Division Two last season and were not given a spot in the top flight after Durham were relegated because of financial issues.
Gillespie, who joined Kent as cover for Allan Donald in March, led Yorkshire to promotion from Division Two in 2012.
"For this whole squad, the world is their oyster," he told BBC Radio Kent.
"Kent have shown that they can do it and they can challenge to play in the first division.
"They play against first division sides in white-ball cricket all the time so the lads know they can do it, it's just a couple of things falling into place."
Kent interim assistant coach Jason Gillespie says they can challenge for promotion to Division One of the County Championship this season.