As a bighead, I had to make an offer (3)
I believe the answer is:
bid
'make an offer' is the definition.
(thesaurus)
I don't understand the rest of the clue.
Can you help me to learn more?
(Other definitions for bid that I've seen before include "Invite" , "Try" , "Offer a price" , "Take part in auction" , "Attempt, offer" .)