The number that was offered initially (3)
I believe the answer is:
two
'the number' is the definition.
(two is a kind of number)
'that was offered initially' is the wordplay.
'initially' suggests taking the first letters.
The first letters of 'that was offered' is 'TWO'.
(Other definitions for two that I've seen before include "Cat's coat" , "Number of faces of the god Janus" , "Number in company, some say" , "Fifth root of thirty-two" , "''... turtle doves''" .)