Increase! The chicken is around almost nine (8)
I believe the answer is:
heighten
'increase' is the definition.
(heightening is a kind of increasing)
'chicken is around almost nine' is the wordplay.
'chicken' becomes 'hen' (hen is a kind of chicken).
'is around' indicates putting letters inside.
'almost nine' becomes 'eight' (eight is one less than nine).
'hen' placed around 'eight' is 'HEIGHTEN'.
'the' is the link.
(Other definitions for heighten that I've seen before include "Make more intense, say awareness" , "Move up" , "Raise" , "Intensify, enhance" , "Intensify; make taller" .)