Absolutely don't signal to anyone higher than you that you might start looking for another job. That's the worst thing you could possibly do and will leave you branded as a lost cause.
What manager is going to stick his neck out or put effort into a subordinate that doesn't even know if they want to be there.
It could also leave you in a compromised position under many circumstances if they wanted to get rid of you / move you etc.
I’ve successfully gotten promotions multiple times at big tech companies after talking to my manager about what other companies are offering and not-that-subtly threatening to leave. The pot is only so big, so sometimes it helps to be the squeaky wheel.
What manager is going to stick his neck out or put effort into a subordinate that doesn't even know if they want to be there.
It could also leave you in a compromised position under many circumstances if they wanted to get rid of you / move you etc.