Hi Paolo
I assume you've already tried the Channel EQ to see if you can isolate and reduce some of the higher bird frequencies.
I would expect that this would have a limited effect and depending on the type of voice, could also start to affect the voice as well.
Have you tried just manually ducking the levels when the subject isn't speaking?
Some things are very limited about what you can fix in post, so you have to be very careful when shooting.
Good luck,
Ben
Curious Turtle Professional Video
Training | Editing |Support
www.curiousturtle.com