Officials representing the Counties are said to be urging Mike Soper to run for the post against Morgan.
Soper declined to comment on whether he had been urged to stand but said: "There is disquiet among the County chairmen over the way things are being handled at Lord's.
"The domestic review group proposals were not popular, particularly as there was only a single representative of the Counties on the group, and we have agreed to set up our own review."
The chairmen are also said to be keen that proposals to streamline the ECB suggested in the Carpenter review are adopted as soon as possible.
Soper added: "The new breed of County chairmen are smart people who know how to run a business and know their sport and many of them would like to see Carpenter adopted swiftly so that the ECB operates more efficiently."