Cabinet member Cllr Malcolm Longley revealed the 'extra pressures coming through from pretty much every departments' during a debate over garden waste collection charges on Tuesday (January 18).
He added it would be a hot topic for discussion at another cabinet meeting, but not a public one.
Opposition finance spokesperson, Councillor Danielle Stone, says the Conservatives needs to come clean.
She said: “The budgetary black hole was entirely avoidable and has been caused by a failure from the Conservative Party both locally and nationally to get to grips with the cost-of-living crisis.
"Life is getting much harder and more expensive for many in our communities meaning more people seek help from the council.
"When these sort of funding shortfalls occur, it is always our most vulnerable that suffer.
"We surely must have expected that the rise in the cost of utilities would also impact our budgets. Why wasn’t that planned in? We are still not seeing the super savings promised by reorganisation. Where are they?
"There is no excuse for these sorts of shortfalls with the technology the council has and particularly with the wealth it hoards.
"Modern technology means we can start from the point every year of asking our services what they need financially to operate and improve. Instead, the council prefers to keep services running on the bare minimum whilst nearly £200million sits in its reserves."
A West Northamptonshire spokesman confirmed public consultations into the draft budget launched last month will continue as planned until February 1 before final proposals go before the full council on February 24.
Labour group deputy leader Councillor Emma Roberts added: “The £6m hole in the council’s budget is nothing short of alarming for many residents who are really struggling to put money in the meter and food on the table.
"The services that the council operate, are crucial in supporting all of those who live in West Northants and £6m is an enormous amount of money to not have accounted for.
"Cllr. Longley justifying this by saying “unknowns do happen” worries me. Historically in Northamptonshire when 'unknowns' happen on Conservative councils it always seems to result in costing residents millions of pounds.
"The draft budget consultation is still open, they should produce the full detail and let the community comment on it."