You may not get the job.

A good manager, absolutely have the chat. There are also managers that would take something like this personally and make it into an issue. e.g what if you are simultaneously looking for internal promotion and some manager decides this makes you look like a flight risk for the new role or any other number of scenarios.

Not every company will blatantly call your former/current employer for a reference, there's procedures for that, and you can certainly control this a bit, as an applicant.

My suggestion would be to openly make this a topic of discussion with your potential new employer. Tell them that you'll have to manage the time as you're still employed and can't just run off during the day easily. They'll understand, even the big ones. I've been able to arrange all interviews at a time I found convenient, and that includes the big US tech giants.