Strauss, who took over managing director duties following the sacking of Ashley Giles, hopes England can find a new coach before this summer's hectic Test schedule gets underway.
The search will be conducted by headhunters SRI Executive, with the successful candidate's first task being the appointment of a new coaching and selection group.
Speaking about the ongoing process, Strauss said: "We are making progress with it and we're obviously very conscious there's a ticking clock, both in terms of appointing a director of cricket and then of course the coach - or potentially coaches - on the back of that.
"The ambition is certainly to have the Test coach in place by that first Test of the summer. In recruitment there are all sorts of moving pieces, including notice periods and I don't think we can categorically say that, but it's the ambition."
If finding a new coach wasn't enough to keep Strauss busy, the 45-year-old is also forging ahead with a high-performance review of English cricket.
The need for an investigation into the structures that underpin the sport became apparent after the Test side floundered in Australia. England's 4-0 defeat in the Ashes series was the culmination of a poor year for the flagship team that gave rise to the term 'red-ball reset'.
Strauss, who is in Barbados with the squad for this week's Test against the West Indies, attempted to flesh out what the concept meant.
He said: "I believe we're looking very strongly at being the best in the world in all formats. If the shop window is functioning well, the knock-on effects for the game as a whole are enormous.
"If we're honest, in Test cricket we have been number one for 12 months in 42 years, with moments where we've been very good and in the top three of four teams. We were in a similar place to white-ball cricket before 2016.
"I sense an appetite for us to be bold in our approach. I think we need to be prepared to be pretty bold and ambitious. I think incremental tweaks is not going to give us the step change we are looking for.
"I've been through two domestic structure review processes before. The problem is if you start with too many constraints, it really limits your ability to think in those bold terms. You've got to start from a blank sheet of paper.
"We want recommendations to be signed off in time for the 2023 domestic season, so that really means by the end of September this year ideally."