The MS Ashdod playmaker reportedly took part in a one-week trial with the Premier League leaders, prompting the Italian boss to snap up the 20-year-old.
Biton, who has nine months left to serve in the Isreal Defence Force, could be granted a work permit, which would relieve him of army duty.
"I have spoken to the Army's top personnel, and that is almost solved," the Daily Mail quotes Biton's agent Dudu Dahan as saying. "City say they can get a work permit for Nir as well, as there is a special category for outstanding prospects like him.
"I have been speaking with Mancini every day, since the trial, so how can I not be optimistic the transfer will be completed?"
Man City are likely to offer a loan deal with the option to buy for £2m.