In a surprise announcement in September last year, Kudva had announced her decision to move on after a seven-year stint.
In response to queries from Business Standard, CRISIL said: "The search process has been completed and we expect an announcement will be made by the end of this month."
It, however, did not comment if the new person would be sourced internally or brought in from outside.
While executive search firm Egon Zehnder has been roped in to find the right candidate, the search process will be monitored by the nomination and remuneration committee of the board of directors.
The committee comprises H N Sinor (chair), Douglas L Peterson, and Vinita Bali. Kudva, too, has been working closely with the committee.
While announcing her intent to step down, Kudva had said: "Change at the top at judicious intervals is a sound governance practice and allows companies to refresh themselves."