Former dual Ashes-winning captain Strauss is an obvious candidate to replace Morris, who resigned to rejoin Glamorgan last month shortly after England had beaten Australia for a third successive time in Test series.
Strauss has admitted that the high-profile situation vacant at the ECB merits his serious consideration, and he is currently giving it just that.
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."
Strauss, 36, retired from all cricket a year ago and passed the England Test captaincy on to Alastair Cook.
He said then he would take his time before plunging into a long-term new career, and has since tried his hand at broadcasting as a Sky Sports commentator during this summer's Ashes.
Cricket administration is, however, another route he has acknowledged may be for him - a suggestion which will chime with many who saw his confident press conferences over the years as an England player and then captain.
"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' position is one the ECB must fill effectively, but also relatively promptly - with the second of back-to-back Ashes set to get under way in November and strategic planning for the next World Cup in 2015 also well under way.
Strauss will nonetheless not be rushing into such a major undertaking.
"They are just suggestions at the moment," he said. "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. In exactly what way that is, kind of remains to be seen a bit."