The Yorkshire coach has been favourite to take over since the departure of Peter Moores, whose second spell in the job ended after the recent tour of the West Indies.
And the ECB’s approach raises the possibility England could go into this summer’s Ashes series with an Australian in charge of the side.
Gillespie is the favourite for the post but admitted this week he would take “some persuading” to quit Headingley, where he delivered last season’s County Championship title under the chairmanship of incoming ECB boss Colin Graves.
The former Test bowler has the backing of England players past and present, with Yorkshire batsman Gary Ballance telling Sky Sports:????”If he was appointed I’m sure he’d do a good job.
“He’s obviously a quality coach. He’s played a lot of international cricket and done very well, and at Yorkshire he’s got the best out of all the players.”
And former England captain Michael Vaughan said: “He’s just a good man, full of energy and the kind of know-how you need at the highest level. If England can persuade him to go for the role I’m pretty sure he’d do a fantastic job.”
Gillespie’s former Aussie team-mate Justin Langer has ruled himself out of the running to replace Moores, whose former assistant Paul Farbrace is in temporary charge going into this week’s first Test against New Zealand.
Farbrace has said it would be “very hard to say no” if he was offered the job full-time.