'authorisable' ultimately derives from English verb 'authorise' (from which the adjective is formed), which comes from Old French 'autoriser', itself from Late Latin 'auctorizare' (based on Latin 'auctor'). In that origin, 'auctor' meant 'originator' or 'promoter' and the Late Latin suffix '-izare'/'-izare' indicated 'to make' or 'to render'.

'authorise' entered English via Old French 'autoriser' and Middle English (as 'authorisen'/'authorise'); the verbal form developed into the Modern English 'authorise', and adding the adjectival suffix '-able' produced 'authorisable'.

Initially related to giving authority or establishing as an originator ('to make someone an authoritative source' or 'to give authority'); over time it shifted to the modern sense of 'to grant permission or official approval', so 'authorisable' now means 'capable of being granted such permission'.