Mills is available as a free agent following his release at Bolton upon the expiration of his contract the Macron Stadium.
A number of Championship clubs are believed to have expressed an interest in the former Leicester City and Reading defender, but Forest are thought to be in pole position to land the centre-half.
Forest boss Dougie Freedman is an admirer of Mills having worked with the player during his time in charge at Bolton, and he feels he would be a valuable addition to his squad next season.
Freedman is the market for defensive reinforcements with Jamaal Lascelles set to link up with Newcastle United this summer and Kelvin Wilson and Jack Hobbs enduring injury-hit campaigns in 2014/15.
Despite operating under a transfer embargo, Forest are able to sign free agents and make loan signings and Freedman is keen to bring in Mills as he looks to build a side capable of challenging for promotion next season.