Offer, British, is declined initially (3)
I believe the answer is:
bid
'offer' is the definition.
(bid is a kind of offer)
'british is declined initially' is the wordplay.
'initially' indicates taking the first letters.
The initial letters of 'british is declined' is 'BID'.
(Other definitions for bid that I've seen before include "Offer; command" , "Offer a price" , "Tender (for work)" , "Make an offer at auction" , "Invite" .)