The fault, caused by a recent core system change within revenues and benefits, equates to more than £10 million in payments.
In reply to a resident confused about why their direct debit due a week before was not taken, Sheffield Council Tax Service said it affected those whose payments were due on May 31st and June 1st.
It said: “We have been made aware of a technical failure within our systems which has resulted in some Direct Debits...failing to collect. We apologise if you have been affected by this and the inconvenience it has caused. We are working to resolve this.
“You do not need to do anything and we will contact you in due course once the issue has been resolved.
“For Direct Debits that were due to be taken on 7th June, we expect these will collect on Tuesday 8th June.”
Councillor Shaffaq Mohammed, leader of Sheffield Liberal Democrats, submitted a question on the topic ahead of a full council meeting next week.
He said: "This has caused a lot of confusion and worry. When a payment doesn’t go through and you can’t get through on the phone, your mind automatically jumps to the worst-case scenario.
“Sheffield City Council need to act swiftly to ensure there will be no financial impact on residents because of the council’s failure."
The council said there are two possible solutions.
One is to take the delayed payment from residents on the 23rd of June and then again at residents' regular payment date of the 30th June or the 1st of July as applicable.
The other option is to take two payments from residents on their next regular payment date due on the 30th of June and the 1st of July.
It said further system testing must be completed before any attempt to take the delayed direct debits and as the council needs to give the relevant notifications under BACS and direct debit rules, the most likely solution is the second option.
Cllr Cate McDonald, executive member for finance and resources at the council, apologised for the incident and said: "As soon as we noticed the issue, we began work immediately to rectify the problem.
"I would like to reassure residents that we are working hard, with all hands on deck, to fix the issue which we hope will be resolved shortly. We will be writing to residents to update them on the situation and inform them of their revised payment date."
She said residents negatively affected by the problem should contact the council's customer services on 0114 273 6633.
Last week, data from the Ministry of Housing, Local Government and Communities showed the council needs to address an estimated shortfall of more than £6.6 million in council tax.
The council faced a budget gap of more than £60 million this year but managed to balance it in March, partly by raising council tax by 4.99 percent.