It comes as the authority increased council tax nearly four per cent across the borough last month in a bid to save £26 million over the next three years.
Bosses have also made drastic cuts including shutting the town’s museum and art gallery and cutting library opening hours.
Opposition leader, Councillor Patrick Harley, blamed Labour for the issue and said the figure was 'shocking'.
He said: “The council is always telling us they have a very good collection rate but to have over £6m outstanding is shocking.
“They should be doing everything they can to get the money.
“They’re cutting front line services year after year but it’d be an easy start to collect the council tax owed because it’s an awful lot of money.”
Iain Newman, who is the chief officer, finance and legal at the council, said: “Like all councils, we do have some council tax owing from a small proportion of our residents, however we do have an excellent record for recovering council tax and our collection rates are amongst the best in the country.”
Opposition councillor Paul Brothwood also hit out at the amount owed to the authority.
He said: “It’s an astromonical amount of money.
“There’s been savage cuts over the past 18 months or longer and the cuts wouldn’t have had to be so savage if we had the money owed.
“It’s peoples duty to pay council tax and if they fail to we have to go after them ruthlessly.
“We want to maximise the money coming in to look after important services.”
The amount owed in council tax is from the period April 1, 2012 to March 31, 2017. The Labour-controlled authority initially put forward a 4.99 per cent increase, but Conservatives and UKIP united to reject the proposal during a meeting lasting almost five hours in March, resulting in a 3.99 per cent rise.
The rise was approved by a narrow margin of 36 votes in favour and 35 against.
The council’s finance chief warned that more cuts would have to be made in the borough after the 4.99 per cent increase was voted down.
Councillor David Sparks said at the time: “Dudley is now in a less secure financial position. We will have to make cuts we wouldn’t have had to.”
The authority collects tax owed through enforcement agents, bankruptcy and insolvency proceedings, deductions from benefits, attachment of earnings and charging orders.