The 26-year-old, who is preparing for this month's African Nations Cup with the Ivorian squad, joined Swansea from Dutch side Vitesse Arnhem for 12 million pounds in 2013.
"As a player it's always good to be part of one of the biggest clubs in the world and it's a good opportunity for me to be in that situation now –- I'm really proud," Bony told City's website.
"I felt excited to wait for this moment and now it's come, it's fantastic."
City have suffered with injuries to strikers Sergio Aguero, Edin Dzeko and Steven Jovetic this season but still closed an eight-point gap to Premier League Chelsea and now trail by two.
With all of City's strikers now fit and Bony signed, manager Manuel Pellegrini has some formidable firepower at his disposal.
In Aguero, Dzeko, Bony and midfielder Yaya Toure, City have four of last season's five leading Premier League scorers.
GOOD ADDITION
"(Bony) will be a very good addition to our squad. We now have four top class strikers and I am looking forward to seeing them work together," Chilean Pellegrini said.
"He is an intelligent footballer who has skill and power."
Bony signed a one-year extension to his Swansea contract in November and the Welsh club's manager Gary Monk said it would take an "astronomical fee" for the club to sell him.
The fee is double Swansea's previous record sale of Joe Allen to Liverpool for 15 million pounds in August 2012.
Bony, the first major signing of the mid-season transfer window, boasts a combination of power, ruthless finishing and quick feet which have made him one of the most prolific and feared forwards in the Premier League.
He scored 25 goals in all competitions in his first season at the Welsh side and has scored nine goals so far this term.
Bony has netted 11 goals in 32 games for the Ivory Coast and will spearhead their challenge at the Nations Cup in Equatorial Guinea which starts on Saturday. If the Ivorians reach the final he will not be seen in City's colours until early February.
(Reporting by Tom Hayward; editing by Martyn Herman)
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