Last year's lowest-ranked council, Nottingham, will see its schools' key stage 2 scores in English and maths climb by a combined total of 20 percentage points.
Newham, bottom of the league in London last year, reported an improvement of 18 points in the proportion of youngsters making the grade in the two tests.
And other urban authorities such as Bristol, Middlesbrough, Knowsley and Portsmouth are among the fastest improvers of 45 councils responding to the survey.
These are the key findings of an analysis of last week's TES survey of provisional key stage 2 results for 1999. This revealed an average improvement of 6 percentage points in English and 11 points in maths in those authorities responding to the survey.
Seven of the eight largest gains reported to The TES are in councils which were ranked in the bottom third two years ago.
This is particularly good news for Education Secretary David Blunkett as his 2002 targets for 11-year-olds will be hard to hit without substantial improvements in the weaker areas, which had appeared to be falling well behind.
Mr Blunkett has promised to resign if 80 per cent of 11-year-olds fail to reach the expected standard (level four) in English and 75 per cent in maths by 2002.
Despite this year's gains, these targets are a challenge for many councils. Nottingham, for example, has to improve by six percentage points a year in maths and five in English. Nationally, English and maths results must rise by four points a year.
Kirklees has reported the biggest improvement so far, at an aggregate 25 point rise, followed by Portsmouth, North-east Lincolnshire, Knowsley, Redcar and Nottingham, all on 22.
Several authorities which were near the top of the tables last year have recorded smaller gains, the lowest being an aggregate improvement of just 8 per cent.
The rapid improvement near the foot of the table is not universal. One inner London authority has only managed an increase of 11 percentage points.
Heather Tomlinson, assistant director (standards and effectiveness) at Nottingham City Council, put the council's improvement down to its reaction to the "shock" of last year's results.
Each school had been given an improvement plan and been visited to identify teaching weaknesses, and 36 schools had been given intensive support.
Ms Tomlinson added: "The first year has really demonstrated that we can achieve great strides in this area."
The official results are expected to be published next week.