The 35-year-old Anelka has already been sent a contract and he is expected to decide on it very soon.
"Talks are at a very advanced stage and if everything falls into place, he will in all likelihood be joining the Mumbai City FC roster soon," a reliable source in the know of the development told PTI.
Sources added that Anelka was yet to "sign on the dotted line".
The former Paris Saint-Germain, Arsenal, Real Madrid and Chelsea striker had been without a club since his controversial exit from West Brom earlier this year.
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Reports also quoted the player's agent Alexander Garini denying speculation that Anelka could have joined Italian Serie A club Fiorentina.
"Fiorentina never made contact with Anelka. Anelka signed a deal with Mumbai FC and so there is no chance of a U-turn. The contract will expire on December 25th, I have it, so there are no problems," Garini was quoted as saying in the media.
Anelka began his career at Paris Saint-Germain, but soon moved to Arsenal. Real Madrid signed him in 1999, before he returned to Paris Saint-Germain. He transferred to Chelsea from Bolton in 2008.
In June 2010, Anelka was omitted by the French Football Federation from the World Cup squad for comments directed against national coach Raymond Domenech.