According to media reports, Barcelona offered Ilkay a 2+1 year contract. Ilkay is 32 years old now. If he plays football in a big club like Barcelona until the age of 35, he will have made a good career planning. After the age of 35, he may have the chance to play football for a few years in Germany or Turkey. It is known that he is a fanatical Galatasaray fan. I think he can play one more year for Manchester City, but if Barcelona's 2+1 year contract offer is true, he should not miss it. First of all, he should make a correct career planning.
I am not saying this to often, but i really think that Ilkay Gündogan is a seriously underrated player. I mean he is the captain of the best Football team of the last few years, which alone is a very great achievement. Of course he is already 32 years old now as you already said, so he won't have that many more years left where he is able to play at his absolute top level. I still think though that is is at least 2 years to early for him to go to a turkish club, because in the end the turkish clubs are second tier at most. He still easily can play 2 or 3 years for an absolute top club.
If Barca is really interested it could make sense for him if he wants to try something new. Then again Barca is a club that is almost broke and still tries to sign expensive players. I think that Gündogan is a kind of player that does not support if a club is acting like that so i doubt he will go to Barca.