You can find the CUCM - Avaya integration documentation here http://www.cisco.com/en/US/solutions/ns340/ns414/ns728/networking_solutions_products_genericcontent0900aecd805b561d.html
Pls rate the post if it helps.
I would definitely recommend SIP integration if you can do this on Avaya, it may need additional hardware and/or license though. From Cisco's perspective if you are going to use SIP I would recommend upgrading CUCM and take advantage of QSIG over SIP which will provide native QSIG features over SIP protocol, i.e. tromboning prevention, etc.