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The 32-year-old has a contract until June 2025 with an option for a further year, while the Spanish champions also said they had included a buyout clause of 400 million euros ($435.72 million).
Gundogan will leave City after captaining the English club to the Premier League, FA Cup and Champions League last season, ending a successful seven-year stint in England where he won 14 trophies, including five league titles.
Barcelona were dominant in every aspect of the game and took advantage of three mistakes by Real Madrid to score.
Barcelona dominated the match with almost 70 per cent of possession and 18 goal attempts to two for Rayo.
Lewandowski, who had 12 months remaining on his Bayern contract, moved to Barcelona in a deal worth 45 million euros and has joined the team on their US pre-season tour.
A sold-out Santiago Bernabeu stadium watched silently as a much-improved Barca under the command of new coach Xavi Hernandez dominated from the start.
Xavi probably could not have asked for much more in his first game managing a struggling Barca team who are sixth in the standings, eight points behind leaders Sevilla.
After the presentation Xavi talked about how discipline will be a major factor in his project as Barcelona manager.
The La Liga club received the green light on Friday to sign the 41-year-old after they paid the release clause in his contract with Qatari side Al Sadd.
Xavi is set to replace caretaker coach Sergi Barjuan who was appointed after the club sacked Koeman.