I think this is very much a similar situation to De Gea to Madrid in 2014.
Bayern need to buy a world class winger in the next 3 months. The obvious choice is Sané, and they're probably the 1 club he'll move to because of his family, but if that move doesn't happen then they will buy someone else.
When that happens, just like how Madrid's need for a keeper went away when they bought Navas, leaving DDG no chance to get to Madrid to be with his girlfriend, Leroy will see his only suitor spend €€€ on someone else, and the gap in their squad won't be open next summer.
In the end De Gea just convinced his girlfriend to move over here and he's been there another 5 years.
Keep Leroy even if he wants to leave this summer, and there's a very good chance Bayern will be off the table and he'd actually rather stay here.
He can't sulk because he has to play to get in Germany's 2020 squad, and every week he doesn't sign, £140k gets away from him forever.