The signings comes less than 24 hours after the club announced the return to West Yorkshire of goalkeeper Casper Ankergren.
The news was quickly followed by the announcement that former Swindon and Millwall midfielder Weston and ex-Coventry centre-back Heath have also committed themselves to United for the next two seasons.
Weston played under Wise at both of his previous clubs, while Heath accepted a new contract after his short-term deal with Leeds expired in June.
The defender is currently nursing a minor injury and missed Saturday's friendly against Wigan Athletic.
Constantine scored 26 times for Port Vale last season but left Vale Park at the end of the term and has been on trial with Leeds throughout the summer.