Gillespie was appointed coach of English county Yorkshire in 2011 and managed to take them from division two to the county championship within just three years.
Clarke played 17 tests with Gillespie, including the fast-bowler's final appearance in 2006 where he scored an unbeaten double century against Bangladesh.
Despite being full of praise for his former teammate, Clarke, as he has done throughout the build-up to the upcoming two-test series against the West Indies, refused to discuss any England team matters.
"Love Diz [Gillespie]. Great fella," Clarke said.
"Unbelievable bloke and I really loved playing with him but I couldn't care less right now who coaches England.
"I'm really focused on trying to beat the West Indies team here.
"I'll worry about England when we have to."
AAP