Shirtliff, Kevin Blackwell's right-hand man, expressed his desire to leave Gigg Lane to new chairman Stewart Day, who only completed a takeover of the financially-stricken outfit at the end of last month.
The Shakers, relegated to League Two, have confirmed they will announce Shirtliff's replacement later on Wednesday.
The former Sheffield Wednesday and Charlton defender joined Bury in June 2011 as assistant to Richie Barker.
He then took the reins on a caretaker basis last August following Barker's switch to Crawley but failed to win any of his nine games in charge before Blackwell was appointed in late September. He returned to assistant.
Shirtliff and Blackwell, however, were unable to prevent relegation from League One.