BCCI treasurer Anirudh Chaudhry and Committee of Administrators' member Diana Edulji have opposed the selection process.
CoA member Diana Edulji had also written to Rai, asking him to put the interview process on hold.
However, the ad-hoc committee formed by the BCCI, comprising Kapil Dev, Anshuman Gaekwad and Shantha Rangaswamy, had interviewed as many as 28 applicants and shortlisted eight candidates in Mumbai.
"Would it not be appropriate to wait till the 17th of January, ie; the next date of hearing before the SC and keep the present matter of coach selection in abeyance till then and have either (Ramesh) Powar continue or have someone else take charge till then?" Chaudhry wrote in his letter.
"It would avoid unnecessary legal issues and unwanted controversy," he added.
India's World Cup-winning former men's coach Gary Kirsten and former opener WV Raman were short-listed for the post along with eight others, including WV Raman, Venkatesh Prasad, Manoj Prabhakar, Trent Johnston, Dimitri Mascarenhas, Brad Hogg and Kalpana Venkatachar.
Chaudhry said the BCCI cannot approve an appointment in the current situation.
"Since there is no decision of the CoA in the matter of the selection of the coach, the process and an appointment if any in the present facts and circumstances, are without jurisdiction and without authority and an appointment that may be a result of this process can in no manner be called an appointment for and on behalf of the BCCI."
Edulji, a former India captain, wrote to Rai on Thursday morning, asking him to put the interview process on hold, but the ex-Comptroller and Auditor General of India gave the go ahead after taking former Supreme Court judge BM Srikrishna's opinion on the legality of the move.
"The email sent by Diana Edulji, member of the CoA, which was copied to the office bearers as well, has touched upon some very fundamental issues. It has given us an unprecedented insight in to the working of the CoA, the professional management and the legal team of the BCCI," Chaudhry wrote.
"I regret to state that it does not paint a rosy picture about the state of affairs with regard to the administration of BCCI. To put it mildly, the decision making seems irregular, the outcome of which may prove to be illegal.
"The execution, blind or otherwise, of those decisions casts serious doubts on either the intention or the competence of those who execute it."