Manchester City president can consider İlkay Gündoğan's stay in Manchester as Manchester City won this season's Champions League title under his leadership. No player would want to join another club while captaining such a consistent and successful team. As his contract runs until the 30th, he still has several days to consider leaving the club. I doubt he will be a regular at all if he joins Barcelona or Arsenal. As he is a regular player and captain of Manchester City, he must consider his position before leaving the club.
With old players like Gundogan, they will prefer to sign new long contracts like 2 or 3 years than only 1-year contract. They want to have as safest contract for their career than short one. Gundogan won many titles with Manchester City and it can be another factor for him to leave the club and find new challenge somewhere else. Being a treble winner is big achievement for Gundogan career and he has nothing to do with Manchester City. Finding a new club and playing at a new league might be more ambitious plan for Gundogan. La Liga and Barcelona are good destinations for Gundogan, a big league, a big club and a matched style for Gundogan.
One of the hardest things about having a player like Gundogan is that he played very well and he even got them some titles, and yet he is old. So what do you do, do you play him even with his further age because he has always provided, or do you send him away and get someone younger? If you keep him then there is a chance he will get worse and you will regret your decision but at the same time fans would love it if you keep him and hate you if he plays badly, it's a huge risk to keep an old player.
However, if you send him away, then you have to find someone that is better or at least good, because if you get someone worse then fans will hate you for sending away a player that provided so many cups for you and played so much for you.