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.
Gundogan is actually the player who was recruited for the first time since Pep Guardiola's first arrival to Man city. So it can be said that Gundogan has helped this club a lot since around 2016 and is still very much needed today.
If I am right, Manchester City bought Gundogan when the player had bad time in his career with some injuries in Dortmund. Pep Guardiola must see big potential from Gundogan to buy him because it is risky to buy a midfielder with a bad injury record. His bet on Gundogan brings sweet fruit in this season but maybe they will both leave the club soon.