Nagaland Governor Ashwini Kumar is said to have conveyed to the Home Ministry his desire to quit the post soon, sources said.
63-year-old Kumar, a retired IPS officer, had served as director of CBI.
West Bengal Governor M K Narayanan is learnt to have sought some time to take a decision on his resignation and conveyed to the government that no pressure should be brought on him during this period.
Narayanan, too, is a former IPS officer who had served as director of Intelligence Bureau.
Before being appointed as governor of West Bengal, 80-year-old Narayanan was the National Security Advisor during the previous Manmohan Singh government.
There have been reports that the government was toying with the idea of allowing CBI to question incumbents Narayanan and Goa Governor B V Wanchoo as witness in the investigations into the AugustaWestland helicopter deal.
The Nagaland governor last week had said he had not received any communication from the Centre but was ready to put in his papers if asked to do so.
"No one has asked me. But if any order comes, I will follow it," he said here after a meeting with Home Minister Rajnath Singh.