I want to know if someone has eventually contacted my manager. I therefore want to send him a formal email to ask him very politely.
Is it correct to write as follow ?
“I would appreciate it if you could inform me anytime/when you could contact my manager.”
Neither of your examples really meet "normal usage".
Both of your examples are really asking for notification of WHEN it would be possible, not for when it has happened.
My choice would be something along the lines of:
I would greatly appreciate it if you would let me know in the eventuality that you contact my current manager.