Healy, 33, had been in discussions with former club Preston but North End boss Graham Westley elected to pull out of the move due to continued ill-feeling from sections of the Deepdale support regarding the player's acrimonious departure to Leeds in 2004.
That cleared the way for north west neighbours Bury to tie up Healy, who is his country's record goalscorer and was a free agent after leaving Rangers at the end of last season.
The Shakers are now awaiting international clearance from Scotland to discover whether their new signing will be available to face Coventry on Saturday in League One.
Healy's arrival follows that of midfielder Shane Byrne, who returned to Gigg Lane from Leicester on Wednesday.
Byrne, 19, made 15 appearances for Bury last season before injury curtailed a loan spell from the King Power Stadium and the player has put pen to paper on a two-year contract.