Tenant not so keen at heart (6)
I believe the answer is:
lessee
'tenant' is the definition.
(I know that lessee is a type of tenant)
'not so keen at heart' is the wordplay.
'not so' becomes 'less' (eg 'not so big' means 'less big').
'at heart' indicates the central letters.
The middle of 'keen' is 'ee'.
'less'+'ee'='LESSEE'
(Other definitions for lessee that I've seen before include "I have to deal with letter" , "Hirer of property" , "One renting a property" , "One holding lease on property" , "One who rents" .)