The London club confirmed on their official website that a deal has been struck between them and their Premier League rivals for the France international.
Nasri trained with the rest of the Gunners squad earlier on Tuesday, but will now not be travelling with them to Italy for Wednesday night's crucial second leg of their Champions League play-off against Udinese.
According to the Arsenal statement: "The move will be subject to Nasri passing a medical and formal registration processes."
That medical is expected to take place in Manchester later on Tuesday, when Nasri will also negotiate his personal terms.
The sale comes after Nasri turned down a new contract offer from Arsenal this summer with just one year to run on his current deal.
The 24-year-old is expected to double his money at Eastlands, where he will join fellow summer signing Sergio Aguero.
Since being taken over by the Abu Dhabi United Group in August 2008, City have spent an unprecedented amount buying top players.
In 2009, they bought Carlos Tevez, Gareth Barry and Emmanuel Adebayor. During the August 2010 transfer window their outlay was around £130 million on players such as Yaya Toure, David Silva and Mario Balotelli.
Under Roberto Mancini, who took over as manager in late 2009, City finished third in the Premier League last season, one place above Arsenal. City also won the FA Cup, their first major trophy in 35 years.
By contrast, Arsenal have already lost their captain Cesc Fabregas to Barcelona this summer in one of the most protracted transfer sagas of all time.
The Gunners have brought in Ivory Coast forward Gervinho from French champions Lille and several young prospects in defender Carl Jenkinson, winger Alex Oxlade-Chamberlain and striker Joel Campbell.
However, manager Arsene Wenger has come under increasing pressure to spend some of the club's significant resources on more experienced players who can help the club compete for silverware sooner rather than later, following six years without winning a major trophy.
Wenger is also suffering from being hit with a two-match touchline ban from UEFA and seeing his team start the Premier League season by taking just one point from a possible six.
England midfielder Jack Wilshere has not travelled to Italy for Wednesday's game, owing to his ongoing ankle injury, with fears growing he could be out for up to two months.
Other players to miss the game include Nicklas Bendtner, Laurent Koscielny, Kieran Gibbs and Sebastien Squillaci. Arsenal lead the tie 1-0.
Arsenal lead Udinese 1-0 after the first leg of their Champions League play-off. City qualified directly for the group stages of the competition.