The authorities, which are not being named, are being closely monitored and given advice to help them produce action plans explaining how they intend to improve standards.
If they fail, they are being warned to expect the same fate as Hackney, which will be reviewed by inspectors from the schools watchdog Ofsted this term and next.
The announcement by schools standards minister Stephen Byers provides more evidence of the Government's willingness to get tough in order to raise standards.
Last month, Mr Byers named 18 failing schools and suggested more could be identified. Mr Byers said they were not confined to urban areas of high social deprivation.
His shot across the bows of under-performing LEAs came as the Government offered education authorities a new deal in helping schools raise standards. Just as central Government set national targets for improving educational performance, so authorities would have a role in setting local targets, monitoring schools' progress in meeting them and offering advice and support where necessary.
Addressing education authority leaders at a London conference yesterday, Mr Byers said the Government was more concerned about Hackney than any other LEA. The authority, the first and only LEA to have a school, Hackney Downs, subjected to a "hit squad" and closed under the last government, had six failing schools and three more showing serious weaknesses, he said.
It had performed badly in GCSE exams and tests for 11-year-olds; a significant number of headships were vacant and there had been no chief education officer for a year.
The LEA, which is controlled by a combination of Conservative and Liberal Democrat members and a group of councillors who have left the Labour Party, proposes that its director of schools should be a relatively junior post, below the chief executive and executive directors.
"What kind of message does that send to the teachers in the borough who are struggling to give children a good education?" said Mr Byers, who said the borough was an example of "lack of direction and drift".
The council's chief executive, Tony Elliston, said the authority welcomed the inspection and would work with the Government.