Yorkshire are making plans for the possible departure of Jason Gillespie to become England coach.
If Gillespie does leave Headingley, the county champions are likely to put director of cricket Martyn Moxon in charge for the rest of the season.
A replacement for Gillespie would then be appointed in the winter.
Moxon could be assisted by former Yorkshire and England all-rounder Anthony McGrath, who works with the club as a player mentor.
He had previously said he would need "persuasion" to take the England job, but his meetings with Strauss have led to the two sharing common ground on how to improve the national team.