Barclay reportedly told ICC directors that he has no intention to run for the chairmanship for the third term.
Shah, a powerful person in the Board of Control for Cricket in India (BCCI), is likely to assume the post in November this year.
The BCCI secretary has the support of cricket boards from England and Australia.
Jagmohan Dalmiah, Sharad Pawar, N Srinivasan and Shashank Manohar are the only four Indians to assume ICC posts.
While Dalmiah and Pawar served as the ICC presidents, the other two were chairmen.