Asserts, avers (6)
I believe the answer is:
claims
I believe this clue is a double definition.
'asserts' is the first definition.
(claiming is a kind of asserting)
'avers' is the second definition.
(claiming is a kind of averring)
(Other definitions for claims that I've seen before include "Declares that one is the owner of something" , "Asserts as a fact" , "Asserts as a right" , "Requests or demands as one's own" , "States that something is the case" .)