The 41-year-old Australian left the White Rose county at the end of last season after winning the County Championship title in 2014 and 2015.
Kent are looking for a stand-in for ex-South Africa paceman Allan Donald, who agreed to join as assistant coach but does not yet have a UK visa.
Ex-player Matt Walker was named the club's new head coach in January.
Donald, who played 72 Tests and 164 one-day internationals for the Proteas, is working towards the England & Wales Cricket Board's level three qualification which is required to gain a visa.
Kent say it could be "weeks or months" before the 50-year-old is able to join up with the squad, hence the search for a coach to fill the role on an interim basis.
Gillespie, who also coaches the Adelaide Strikers in Australia's Big Bash Twenty20 competition, joined Yorkshire in 2011 and led them to promotion to the top flight in 2012.
He held talks with England's director of cricket Andrew Strauss in May 2015 over replacing the sacked Peter Moores as national coach, before the role eventually went to fellow Australian Trevor Bayliss.
Kent finished second in the Division Two table last season and missed out on promotion, with Jimmy Adams departing as head coach at the end of the campaign after five years in the post.
Former Yorkshire coach Jason Gillespie is in talks over becoming interim assistant coach of Kent.