When to give notice? Contract signed, but third-party background screening not started yet

I’ve recently accepted a job offer and signed the employment contract. The next step is background screening through a third-party pre-employment checking provider. Notice is 3 months.

​I’m stuck on when to officially hand in my resignation to my current employer:

​Option A: Give notice now. If I don't give notice, the screening company will contact my HR/manager for a reference before I’ve told them I’m leaving, which isn't ideal.

​Option B: Wait until screening clears. There’s always a tiny residual risk of an offer falling through or being withdrawn during background checks, so handing in notice before full clearance feels risky. And my manager could know I'm leaving from that third party company asking for references.

​ Does the screening provider typically ask before contacting your current employer, or do people usually hand in their notice as soon as checks start?