Garcia, 25, is expected to fill the gap left by De Jong, who joined AC Milan earlier in the day, although he can also play in the centre of the defence.
The Spaniard was a regular at Benfica after arriving from Real Madrid in 2009, scoring six times in 73 appearances for the Portuguese club.
Garcia, who has one cap for Spain, will wear the number 14 shirt at City.
Brazil right-back Maicon signed from Serie A side Inter Milan for an undisclosed fee.
"Roberto Mancini is an admirer of the 31-year-old having worked with him for two seasons while in charge at Inter and the defender is familiar with the 3-5-2 formation the City boss has been experimenting with in pre-season," the club said on their official website.
Maicon played 235 matches for Inter, scoring 20 goals, and has previously played for Cruzeiro and Monaco and has won 66 caps for his country.
He will link up with his new team mates when he returns from international duty next week.
Mancini said: "We are talking about one top player in Maicon, because he has experience."
Sinclair also arrived for an undisclosed fee from Premier League rivals Swansea.
The 23-year-old winger was a member of the Britain team at the London Olympics.
"I'm just glad everything is settled and that I'm now a City player," Sinclair told the website.
"There were times when I thought it might not happen so I'm relieved that I'm here and I can't wait to get started."
Sinclair could be included in the City squad for Saturday's home game against Queens Park Rangers.
Wright joins the club as third-string goalkeeper behind Joe Hart and Costel Pantilimon.
He was a free agent after leaving League One club Preston North End.
Mancini said: "We need a goalkeeper with experience and Richard Wright is good for us."
City later announced the signing of teenage Serbia defender Nastasic from Fiorentina for an undisclosed fee.
The 19-year-old joins in a deal which also sees Montenegro international Savic moving in the opposite direction.
Nastasic will arrive as a straight swap for Savic, who has struggled to make an impression at the Etihad Stadium since his £6million signing from Partizan Belgrade last year.
He joins captain Vincent Kompany, England international Joleon Lescott and Kolo Toure among manager Roberto Mancini's central defensive options.
Dutch midfielder de Jong has been included in the AC Milan squad to face Bologna after joining the Serie A side on a three-year-contract City.
"Nigel De Jong, much to the satisfaction of the Milan faithful, is now one of our players," the Italian club said,
The 27-year-old joined Manchester City in 2009 and made 104 league appearances for the club.
"The signing of De Jong and Spain winger Bojan Krkic, who had been on loan at AS Roma from Barcelona, meant Milan should now challenge for the title, vice president Adriano Galliani said.
Malaga signed Paraguay striker Santa Cruz on a season-long loan.
The 31-year-old was on loan at La Liga side Real Betis last season.
- Roberto Mancini
- AC Milan