The Bundesliga champion was due to begin its cup campaign on Saturday against the fifth-tier side.
However, Bremer recorded positive COVID-19 test results within the first-team squad, meaning quarantine measures have been imposed on the club.
ℹ️ The DFB has announced that our #DFBPokal match at Bremer SV on Friday has been postponed due to quarantine measures ordered by the authorities for the host club.— FC Bayern English (@FCBayernEN) August 3, 2021
"The health of our and of our opponent's players are of paramount importance, so we consider the order of the health department to place our team in quarantine as correct and necessary," said Dr. Peter Warnecke, the club's first chairman and hygiene officer.
Bayern later said in a statement: "The DFB [German Football Association] has announced that Bayern's DFB-Pokal match at Bremer SV on Friday has been postponed due to quarantine measures ordered by the authorities for the host club.
"The DFB will make a decision on the rescheduling of the match after consultation with both clubs as soon as possible. In addition, the draw for the second round will also have to be postponed due to both clubs having been placed in different sides of the draw."
Bayern later announced on Tuesday that young forward Joshua Zirkzee will spend the 2021-22 season on loan with Anderlecht.
Sporting director Hasan Salihamidzic said: "It's very important for Joshua that he gets as much playing time as possible. Anderlecht offers him this chance. Joshua can now take the important next step in his professional career in an interesting league, and I trust him to have a good season."
Bayern is due to begin its Bundesliga title defence away to Borussia Monchengladbach on 14 August (AEST).