Two police forces in the north east of England have expressed concern over budget cuts totalling £2.6m.
The Home Office wants forces in England and Wales to cut 1.5% of their budget - a total of £125m - to help tackle the national deficit.
Durham Police has described its £1.3m cut as "a blow" and said it would be hard to make further reductions.
Cleveland Police, facing a loss of the same amount, agreed that it would be hard to implement.
Dave McCluckie, chairman of the Cleveland Police Authority, said: "This is a very, very difficult cut.
"It has been made in a year in which we've already formed and balanced our budgets, and made or started the necessary investments which were to take us through the year."
Peter Thompson, who chairs Durham Police Authority, said: "We've been very austere for several years and have probably saved about £10m to address needs.
"But now, all of a sudden, and not too long into the new financial year we've been requested to find another £1.3m which is going to be very difficult for us."
He added that he had earlier pledged there would be no reduction in front-line policing, but could not guarantee there would be no job losses elsewhere.
"If you are looking at cuts of this nature so quickly, clearly it will affect jobs," he said.
"We will be looking at ways and means of downsizing our operations in the future to effect the saving."