The former Australia fast bowler has been linked with the role since Peter Moores was sacked over the weekend.
His international team-mate Justin Langer, currently with Western Australia, and former Sri Lanka coach Tom Moody have also been talked of as possible candidates.
Strauss only started work last week, and finding a replacement for Moores will now be his priority with an Ashes series only a couple of months away. Assistant coach Paul Farbrace will be in charge for the two-Test series against New Zealand, whih starts next week at Lord’s??
“He’s one of the candidates, absolutely, ” Strauss said when asked by Sky Sports‘ Tim Abraham if Gillespie was top of England’s shortlist.
“At the moment I have just been concerning myself with the process and how we go about it. Over the next few days we’ll start approaching candidates.
“I need to go and meet them, speak to them and see their philosophy on cricket, their philosophy on coaching and whether it’s aligned with what I think is in the best interests of England cricket.
“I think there will be one over-arching coach (for Tests and one-day cricket). I think you need one person in control and in charge of that environment.
“Also it’s important for people to know I am not going to be playing a tracksuited role. The execution of strategy is over to the captain and coach as it always has been. It’s the right way to go. My job is to plan for the future.”
Strauss also revealed how tough it was to make his first job as director of cricket the removal of Moores, who he had previously played as captain under.
“It was a very difficult decision for us to make,” Strauss added. “Peter Moores has been very popular in the England dressing room.
“He’s got a very good record of developing players but I felt that in some areas of international cricket he is a little bit exposed, around tactics and strategy. I think we saw that in the World Cup and also he was undermined by the fact he had done the job before.
“We got to the situation where every game was a referendum whether the coach should stay or not. That’s not fair on him or on the players. I felt it was the right call to make and my job now is to find the right coach to take England cricket forward. I’ve got a lot of work to do in the coming days and weeks to make sure that happens.