The England international - normally used at right-back - is said to be in talks with the Blues over a five-year contract and a deal is expected to be completed within the next few days.
Mills has been looking to quit Elland Road since going on loan to Middlesbrough last term and has been allowed to leave for free.
Boro were hoping to make his move permanent but cooled thier interest due to Mills' wage demands - around £2m a year - which have also put off Birmingham, Newcastle and Tottenham.
Should he sign, Mills will become City's third summer signing following the arrival of goalkeeper Geert de Vlieger and left-back Ben Thatcher.