Adams left the Shakers when his contract ended during the summer and signed for the Foxes despite the two clubs failing to agree a transfer fee.
Because he was under 23 at the time of the move, the League One club were forced to pay a sum which has now been set by a tribunal.
Leicester must pay £100,000 immediately with a further £25,000 when the player has made 15 appearances for the club.
The total fee for the midfielder, who has made three appearances for his new club, could rise to £225,000 subject to further clauses being met.
"We think that is a fair result and we are fairly satisfied with the outcome," the Bury director Ian Harrop said.
"Clubs have to realise that if you take a chance on taking our players you are going to have to pay for them.
"I said to Nicky after the tribunal we wish him the very best. He has to get into that first team and stay there."