Councillor Graham Ballinger, who leads Wyre Forest District Council, made the plea amid the authority's financial position "getting worse".
It comes as the council now faces an estimated £2.7 million shortfall per year – with the finances taking a blow due to the impact of coronavirus.
Councillor Ballinger said they would "consider" the increase – either £8 or £10 for Band D properties – in order to keep vital services running.
But he stressed a consultation would be launched for residents and businesses to have their say on the idea later this month and during early December.
More Covid-19 coverage:
He added: "Once again we are faced with very difficult decisions. The council's financial position is getting worse and we need to implement big changes to close the budget gap.
"Wyre Forest District Council collects council tax but the majority goes to other organisations. We keep only around 12 per cent of it. This helps to fund providing our services.
"Our proportion of council tax has gone up by only 11 per cent in the past decade. This is well below the rate of inflation, and in real terms it has fallen.
"We hope that the Government will recognize the financial plight facing this authority and virtually all other councils. We would like greater choice about council tax increases in the future.
"If we are given more flexibility we would consider increasing our proportion by £8 or £10 at Band D, which would be about 40p to 50p a week.
"For most people it would be a lot less than this because most households fall into Band B. We will be asking residents' views on whether they would be willing to pay this small extra amount to keep essential local services going."
Wyre Forest District Council currently receives a total of 12 per cent – 12p in every pound – of council tax they collect from people in the area.
Other ideas will be discussed at a meeting on Tuesday – with a business case being commissioned on Wyre Forest House to examine the financial impact of disposing some or all of it.
They will also consider ending the community leadership fund which gives each councillor £1,000 a year to support local organization and activities.
And the number of councillors could be reduced – with the council potentially asking the Boundary Commission for a review to cut the numbers from 2027.
Council chiefs say the authority has incurred additional costs due to coronavirus and a significant drop in income – with the costs not being fully covered by the Government.
The proposals will be discussed by the cabinet meeting at 6pm on Tuesday with the survey going live on the council website the following day.
Recommendations from the meeting and feedback from the consultation will be brought to the cabinet meeting on December 22 as part of the full Medium Term Financial Strategy Report.