Councils had been told that 78 per cent of their 11-year-olds should reach the expected level in the tests. But just 28 of the 150 English education authorities met that target last year in maths and 66 did so in English.
Ministers have now told councils that the target will be dropped. They said they recognised that for some authorities, reaching 78 per cent would require "a very significant increase above their current position".
Provisional results for the City of Bristol, which was in the bottom 10 authorities last year, show that 72 per cent of pupils achieved the expected level 4 in English and 68 per cent in maths.
Results in the London borough of Barnet, which last year was the fourth-best performing authority, remain unchanged with 84 per cent of pupils reaching level 4 in English and 81 per cent doing so in maths.
Key stage 2 test results are expected to show a rise again this year, when they are published on August 23.
Results improved dramatically in the first few years after Labour came to power before levelling off between 2000 and 2003.
Last year saw an increase of 3 per cent in the proportion of primary pupils reaching the level expected of their age group in English and 1 per cent in maths.
A TES survey in July which covered 3,700 pupils predicted a rise in the percentage of pupils hitting the expected level of 1 point in English and 2 points in maths. This would mean 79 per cent of pupils reaching the expected level in English and 76 per cent in maths.
The national targets - which say 85 per cent of all pupils in England should reach the expected level in both subjects - remain in place.
The targets were originally set to be achieved by 2004, but have been moved to 2006 and are expected to remain at least at that level until 2008.