Strauss, a former dual Ashes-winning captain, is a candidate to take the place of Morris, who last month resigned to rejoin Glamorgan shortly after England beat Australia for a third consecutive time in the Test series.
Strauss said that he will seriously consider the job, and price per head services wonder what he will ultimately decide to do.
He said: “One of the things you’ve got to be very careful with when jobs like these become available is to understand the actual job properly.
“I’ve obviously seen Hugh Morris do parts of his job, but not the whole job.
“I’m looking into it, but they’re decisions that need to made over the next couple of weeks, and I’ll think quite long and hard about whether it’s the right time and the right job for me.”
Pay per head sportsbooks saw that a year ago, Strauss, 36-years-old, retired from all forms of cricket and gave Alastair Cook the England Test captaincy.
Then he said that he would take his time before jumping into a long-term new career in the sport, and during this summer’s Ashes he has since tried broadcasting as a commentator for Sky Sports.
But cricket administration is another path that he has acknowledged may suit him – a suggestion which will go very well with pay per head sportsbooks.
“For me, I haven’t decided one way or another whether that’s the right job or the right time for me yet,” Strauss added.
“But I’m certainly looking at it, as will a lot of other people be.”
Morris’ role is one that must be effectively and quickly filled by the ECB, as the second of back-to-back Ashes are set to get underway in November, and plans are also underway for the next World Cup in 2015.
But Strauss will not hurry into such a major responsibility.
“They are just suggestions at the moment. There’s obviously an application process for that MD of England cricket job. No doubt there are going to be a lot of good candidates looking to do the job.
“I’m always looking for opportunities and I will continue doing so. I really enjoyed doing the Sky stuff. I’m very passionate about the game of cricket and want to contribute to it.
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“In exactly what way that is, kind of remains to be seen a bit.”