Sven-Goran Eriksson's Foxes now look favourites to land centre-half Mills, who has already been the subject of a failed bid from Wolves.
Stoke City are also known to be keen on the former Manchester City trainee, but the Foxes are hoping to steal a march on their rivals.
Leicester had an initial bid of ?3.75million turned down by Reading, but an increased offer has met with the Royals' approval.
Mills is now set to discuss personal terms with the Foxes regarding the possibility of joining Eriksson's side for next season.
Eriksson has been looking to construct a squad capable of challenging for promotion to the Premier League and has already signed the likes of Kasper Schmeichel, Lee Peltier, Sean St Ledger and David Nugent this summer.
The Leicester boss missed out on Craig Mackail-Smith earlier in the week and will hope for better fortune in trying to persuade Mills to move to the King Power Stadium.
The 24-year-old joined Reading from Doncaster Rovers two years ago and he has made over 70 appearances during his time with the club.
Source: Team Talk
Source: Team Talk