Desai was questioned on Friday, Apr 23 by Central Investigation Bureau (CBI) sleuths.
"Investigations are on, but we can't provide further details," CBI spokesman Harsh Bahl told a news agency.
The two other people arrested by the CBI have been identified as Jitendar Pal Singh and Kanwaljit Singh.
The CBI reportedly caught the MCI president red handed when he was accepting the money.
The function of the Medical Council of India is to establish uniform standards of higher qualifications in medicine and recognition of medical qualifications in India and abroad.
Even in the past, Desai has been accused of granting recognition to many institutions which did not meet the qualification criteria.