The 40-year-old Gillespie, who took 259 wickets in 71 tests, has coached English county Yorkshire for the last four years, leading them to the championship title in 2014.
"I've spoken to (England cricket director) Andrew (Strauss) and the ECB about the vacant position," Gillespie told Sky Sports.
"We had a good chat and we'll see what pans out from there."
Peter Moores was sacked as England coach two weeks ago after a disappointing year at the helm and Paul Farbrace was placed in charge of the team for the ongoing two-test series against New Zealand.
"They made it very clear they're chatting to a number of people of interest, so they'll let us know what the process is in due course," Gillespie said.
Gillespie was installed as one of the favourites to get the England job along with former Australia opening batsman Justin Langer, who ruled himself out by re-committing to coaching Western Australia.
(Reporting by Ed Osmond, editing by Pritha Sarkar)
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