Gillespie is the clear favourite to replace Peter Moores and could be appointed in the next few days.
Gillespie travelled to London on Thursday to meet Andrew Strauss, the new England director of cricket, and Tom Harrison, chief executive of the England and Wales Cricket Board.
The former Australia fast bowler described the meeting as positive and it is thought that personal terms have already been discussed.
Speaking at Trent Bridge on Friday evening, ahead of Yorkshire’s NatWest T20 Blast fixture against Nottinghamshire, first-team coach Gillespie broke his silence on speculation linking him to the national post.
Asked whether he had had any contact from the ECB about the coach’s job, Gillespie said: “Yes, I have.
“I did go down to London yesterday (Thursday) to chat with Andrew and Tom. It was a really good chat.
“It was made very clear to me that there are a number of people that the ECB are talking to about the role, and they’ve got a process they need to go through.
“Until I hear anything else, I’m just going to continue on as normal.”
Although the ECB are speaking to other candidates, Gillespie is the clear frontrunner.
The 40-year-old is expected to be in place in time for the Ashes series starting in early July, when he will come up against his great friend and former team-mate Darren Lehmann, the Australia coach.