Highly placed sources said that other names being considered include RBI Deputy Governor Urjit Patel, former Deputy Governor Rakesh Mohan and Chief Economic Advisor Arvind Subramanian.
"It (announcement of new Governor) will happen reasonably in advance. We don't want unnecessary speculation," said a top government source, adding that "the process of selection is already on".
With Rajan's term ending on September 4, sources said the government would like to announce the next Governor of the Reserve Bank of India by July end.
Interestingly, some of the contenders whose names are doing the rounds at the Finance Ministry, including Patel and Subramanian, have worked at IMF - same as Rajan.
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Besides, Mohan has been the representative of India as Executive Director at the International Monetary Fund, a post currently held by Gokarn.
Bhattacharya, sources further said, is unlikely to get an extension as chairperson of State Bank of India, indicating that she could be considered for the top job in the Reserve Bank. Her three-year term ends in September.
Buffeted by political attacks coupled with unending speculation over his continuance, Rajan on Saturday announced that he would return to academia after expiry of his term on September 4, putting to rest all speculation over the issue.
"...On due reflection, and after consultation with the government, I want to share with you that I will be returning to academia when my term as Governor ends on September 4, 2016," Rajan had said in a message to the RBI staff.
Patel joined the RBI as Deputy Governor on January 11, 2013 and was given a second term in January this year. He has been the key architect of inflation target and rate-setting panel.
A close lieutenant to Rajan, his appointment is widely seen as ensuring continuity in policy and reforms initiated since 2013.
Patel's appointment comes at a time when inflation has breached the 6 per cent top range set by the government. He would also have to follow through on Rajan's efforts to clean-up NPAs at banks.
He was with International Monetary Fund (IMF) between 1990 and 1995 and worked on the US, India, Bahamas and Myanmar desks.
Banking sector reforms, halving inflation rate and getting government to target inflation at 4 per cent were other achievments of Rajan.
The renowned economist had also won plaudits for helping arrest the rupee's decline against the dollar and promote price stability but his focus on arresting inflation by not cutting interest rates also led to bitter criticism from politicians particularly BJP MP Subramanian Swamy.
Just as Patel's announcement was made, Swamy took potshots at Rajan for allegedly continuing to hold US Green Card.
RBI had in June stated that Rajan is an Indian national and an Indian passport holder.
When another follower said if Patel will be hawkish as Rajan, Swamy tweeted: "Don't be an idiot like the presstitutes."
Patel is the eighth Deputy Governor in Reserve Bank's history to be made its Governor and the sixth with an experience at International Monetary Fund (IMF).
Later, he was on deputation (1996-1997) from the IMF to the Reserve Bank of India, before becoming a Consultant (1998 -2001) to the Ministry of Finance.
Incidentally, it was a Saturday when Rajan surprised everyone on June 18 about his decision against a second term, while the announcement for Patel's appointment also came on a Saturday.
Rajan was also appointed for a period of three years, which has been the case for most Governors so far.
Patel is the eighth Deputy Governor at RBI to be made Governor, the last being Y V Reddy. Besides, at least five former Governors had served at IMF before becoming RBI chief, while a few others went on to work at IMF after leaving the central bank.