Kavanagh, who had been linked with a temporary switch to Elland Road, has now joined Sheffield Wednesday on loan until the end of October.
Wise was happy to miss out on the Republic of Ireland international and admitted he has no immediate plans to increase his squad.
United's boss was expected to use the Football League's emergency loan market to strengthen his pool of players having seen his summer transfer plans wrecked by the club's spell in administration, and an additional striker was seen as a priority last month after Leon Constantine and Tore Andre Flo were sidelined by injuries.
Kavanagh has made more than 400 league appearances and played a central role in Sunderland's promotion to the Premiership last season, but Wise insisted the 33-year-old would not have suited the young squad he is constructing at Elland Road.