Family HSA for newborn.

I have a newborn. I have a high deductible medical HSA plan from work. Can I claim family HSA amount if I enroll my child into my high deductible plan under two scenarios:

Scenario 1: I claim my child as dependent on my taxes
Scenario 2: my partner (who is also child’s father) claims the child as dependent on his taxes.

Please note that my partner and I  have separate medical insurance (different employers)

thank you