In a personal piece penned by Karan Thapar, he talks of 'unimpeachable' sources that claim that Raghuram Rajan was indeed offered a two-year extension as the Governor of the RBI, but the offer was turned down. The sticking point turned out to be Rajan's inability to get a further extension of leave of absence from Chicago University without losing his valuable tenure.
While Rajan offered to stay for a further 8 months, this was deemed to short by the ruling BJP government that wanted a long-term solution for the stake of financial stability.
Thapar also emphasizes his position as a staunch Modi critic, and reiterates that the statement was issued as a way to lighten his own concience, rather than representing a shift in his allegiances.
He also reveals that Rakesh Mohan was first offered the post after Rajan's denial, but the erstwhile Deputy Governor of the RBI declined the offer for personal reasons. It was only at this point that Urijit Singh was considered for the job; one that he eventually accepted.
You can read the original statement from Karan Thapar here:
http://www.hindustantimes.com/columns/the-modi-government-does-not-deserve-the-flak-it-got-for-raghuram-rajan-leaving-the-rbi/story-COSWUEpappcdyPuueYhuZJ.html