It had been expected that a final pitch inspection would take place on Tuesday morning, but Bury have revealed that they "were fighting a losing battle".
"An earlier-than-planned pitch inspection has ruled out Tuesday's game and as such it has been postponed. The inspection was brought forward to today as it became more and more obvious during the day that we were fighting a losing battle," the statement from Bury read.
"Large areas of the pitch remain frozen solid and with snow falling throughout the day, as well as falling temperatures, it became evident that there was no way this game could have been played.
"With full agreement from the Football League, Tuesdays night's game has now been postponed. A new date will be announced once confirmed."
Bury currently occupy 22nd spot in the League One table, while Sheffield United sit in fifth position.