Morris has confirmed he will be leaving the role to become chief executive of his former county Glamorgan, opening up an attractive vacancy within the England and Wales Cricket Board.
Strauss, who only retired as a player last year, earned huge respect during his time as England captain and fits the bill as a highly intelligent, statesmanlike figure in the game.
He has also done some work with the ECB since the end of his playing days, although this summer he also dipped his toe into the water of television commentary.
Morris, who would be willing to assist in the search for his replacement, sees no reason why Strauss could not take his mantle.
"I've not spoken to Andrew about it because I only handed my resignation in on Sunday, but he is an incredibly capable man," he told Press Association Sport.
"He is a really likeable man, he has fantastic skills and the knowledge and experience of the game at all levels.
"If he wanted to do it and if he got the job he'd be great but I'm sure there will be a lot of good candidates.
"I'd be happy to talk to the board (about the recruitment process) and the one thing I can say is it's a fantastic job.
"Whoever is going to sit behind this desk in future has a terrific job."
Another prominent member of the England set-up who may yet be interested is team director Andy Flower.
He is committed to leading the side in this winter's Ashes Down Under, but has yet to publicly commit himself beyond that point.
An office job at Lord's would be a significant shift in pace for Flower, but he is known to find the workload and travel commitments a drain.
Morris believes he is another with the credentials to succeed.
He said: "Both Andy Flower and Andrew Strauss are fantastic guys, I know them both well and I worked with them for a long time.
"Whatever job either of them wants to do they will do very well, no doubt about it.
"If they fancied this job, or applied, either of them would be great but that is for others to decide."
Morris admitted he would be exiting his current role with a heavy heart, probably in January, but is excited about the road ahead.
"I've really enjoyed the last 16 years at the ECB, and the last six in particular," he said.
"There have been lots of challenges along the way, but when you get successes like we have achieved this summer it is very satisfying,
"Of course I will miss it, but I'm very lucky to have the chance to move back to Glamorgan.
"I'm really looking forward to working with the players and the staff and putting a smile on their faces."