And a player who might well be the bridge from where they are (fledgling) to where they desperately need to be: grown-up, tougher and smarter.
Had he wanted to, the 32-year-old could easily have multiplied his new salary by 10 thanks to the Saudia Arabia Public Investment Fund splurge. He chose not to.
Like his old Borussia Dortmund teammate Robert Lewandowski, whose goals effectively turned Barcelona into a title-winning team last season, Gundogan was seduced by the persuasive effervescence of coach Xavi Hernandez, by the beautiful life in the city where his new club is based and by the romantic idea of revitalising a footballing identity that remains faded and jaded.
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Unlike Lewandowski, who'll almost certainly score more goals and look far more comfortable with Gundogan supplying him, Barcelona's newest recruit already felt genuine affection and affiliation to the club -- or at least his teenage impression of it.
When I was at the City's Etihad Campus in the week of the recent Champions League final, interviewing Erling Haaland, I bumped into Gundogan -- only for enough time to congratulate him on a scintillating season and wish him success against Inter in Istanbul.
Before his previous Champions League final (the defeat to Chelsea two seasons ago), however, I spent an hour in Gundogan's company.
It was then, probing for anecdotes to better understand him, that he shared some of the feelings that have since motivated him to turn down a contract extension with the treble winners and to reject the idea of moving -- with Cristiano Ronaldo, with Karim Benzema, with N'Golo Kante -- to play in Jeddah or Riyadh.
Gundogan told me: "When I was a teenager ... I admired Pep Guardiola's Barcelona side so much for how they would play.
"Not just the success and lifting all those trophies, the way they were playing. When Pep was in charge of Barca, my biggest dream ever was just to play in that side.
"It was so far away, but it was incredible to watch. I just loved it so, so much.
"When Pep came to the Bundesliga, it was exciting, and it was so, so difficult to play against his [Bayern] side. They always had a plan, and their plan was always better than your plan."
Guardiola, at Man City, became such a guru for Gundogan, such a messianic figure, that it's pretty natural to want to play for a man of similar (if not identical) ideas -- an evangelical disciple of the brand of football Gundogan loves.
That man is Xavi. It has become interesting, across that LaLiga-winning season, to hear the Barcelona manager and a handful of his new players talk about one another.
Meaning that the process wasn't him identifying a type of footballer then settling for whoever the club found was most available. Xavi targeted specific footballers. And got them.
Then, when Lewandowski, Christensen, Kounde, Raphinha and Franck Kessie have spoken about their reasons for joining, sometimes an uncomfortable decision while they waited, sweating, to see when or indeed whether they could be registered to play because of Barcelona's horrible financial fair play situation, every single one of them has said: When Xavi spoke to me, I was convinced.
So it had been with Gundogan.
Barcelona, in the shape of football directors Jordi Cruyff and Mateu Alemany, were working on the mechanics of this move for many months. But it's the immediate mutual feeling between the two diminutive midfielders, the 43-year-old Catalan treble winner and the 32-year-old German treble winner, that properly sealed the deal.
Gundogan plays, and thinks, the way that Xavi desperately needs if he is to convert the pragmatic, often defence-oriented, prosaic football that won him Spain's title in his first full season into a brand of soccer that can, at least, make Barcelona competitive with Europe's elite. Which their recent ties with Bayern Munich, Internazionale, Eintracht Frankfurt, Manchester United, Juventus and Paris Saint-Germain have shown that, categorically, they are not.
Again, there's a passage from my interview with the likeable, articulate and deeply impressive Gundogan that I think (notwithstanding his brilliance for City) sheds light on what he, as a man, brings to this stage of Barcelona's evolution. He talked to me about how much he adored and lusted after the Champions League trophy but pointed out that his beginnings in the tournament were brutally rough.
Coached by a subsequent Champions League winner (Jurgen Klopp), playing with the man who would score the winner in a World Cup final (Mario Gotze) and playing with another future treble winner (Lewandowski) among many other fine footballers, Gundogan tasted defeat in his first two Champions League matches, against Marseille and Olympiakos. Then he was dropped.
Read his words extremely closely then try telling me they don't remind you of the naive, fragile way Barcelona have played in recent UEFA competitions.
"I remember my first season in the Champions League with a talented Dortmund side," he said. "Everyone was expecting us to go far.
"It just hit us like a bus. It was so difficult to anticipate how to deal with it. We did really badly in that first season in the UCL with Dortmund.
"No blame. No regret. We were just not ready. Sometimes that's the case.
"Sometimes you need experience, to smell, to feel, how it is. To be on the pitch, flying away, staying over a couple of nights, playing in the evening in front of crowds which are incredible and stadiums which are full of emotions. We had to adapt and go through that frustration."
That describes Barcelona's past three calamitous European campaigns perfectly.
Gundogan adapted and played with such magical, elusive distinction that he went to the Champions League final with Dortmund, with Man City two years ago, won the treble this season (where he played an important role in Rodri's winning goal) and would have been a World Cup winner with Germany in 2014 but for a horrendous back injury.
His role, now, is to add brains, experience, know-how, technique, skill and seniority to a Barcelona midfield that, even before Sergio Busquets left, simply does not use the ball effectively enough nor screen the defence sufficiently well.
Additionally, one of Gundogan's many values now is to play sufficiently often in left midfield that Xavi, in fact the whole Barcelona side, isn't so fatally dependent on Pedri that he plays and plays and plays and plays until he is injured then misses months.
Gundogan is multilingual, terrifically bright, polite, interesting and humble ... but do not, at any cost, mistake him for someone who isn't tungsten tough. He will make it his job to add the mental and emotional steel that Frenkie de Jong, a talented player, desperately lacks.
Gundogan will teach by word, by deed and by training performance. He will be Xavi's lieutenant on the pitch. Watch out for Gundogan lambasting players who make bad decisions, who give the ball away, who are naive positionally.
His view is: "Adversity develops you as a person not just a footballer. The whole journey itself which comes with this career, my general and social life -- I appreciate it.
"The sad bits and the setbacks and the difficult moments. I had a lot: defeats, injuries. But always I was able to react, to come back well.
"Every time it gave me satisfaction that I was strong enough to come back well and be again on top of my game. I think that's the challenge of life in general, not just football.
"Always you do your best; sometimes it's not enough ... you lose, you fail. But then it's about standing up again and trying to proceed again."
If you watched Barcelona play last season, don't those sentiments look like just what Xavi is trying, and partially succeeding, to inculcate?
You've watched Gundogan star for club and country. You need very little reminder of his agility, his technical excellence, his movement, his increasingly confident finishing or his will to win and win and win.
But perhaps these words of his give some insight into who he is, what he believes and how he functions as well as some understanding of why Xavi wanted to sign him with the same intense determination as Guardiola once did, nudging him in the tunnel before a Bayern vs. Dortmund match and warning him that, one day, they would work together. They did, and they made beautiful music together.
This task, in a Barcelona side that is uneven, youthful, dislodged from its Camp Nou home and facing a significantly more dangerous challenge from Real Madrid, looks more difficult and more demanding. But Gundogan is special, as a man and a footballer. He might be precisely what Spain's champions need.
And, at any rate, he'll be damn good fun to watch.