Clare Griffiths was offered �2,000 by a desperate parent

Some 70% said parents had lied in applications, for example giving a false address to qualify for the school's catchment area.

One family in Hampshire had listed an empty field as their home, the Six O'Clock News survey revealed.

Researchers spoke to the heads of 50 of England's 100 leading primaries.

'You gain and I gain'

Most said the pressure from parents was increasing, with 32% having been offered financial bribes, other inducements, or even receiving threats.

The head teacher of Parkview primary in Derby, Clare Griffiths, said: "A gentleman in the first year the school opened came in to the school and said 'it's a new school, you're building up your resources.

"'If you will give my child a place you can have �2,000 to add to your library resources. You gain, and I gain.'"

The parent showed surprise when told this was not possible.

On a separate occasion, the school alerted the police after a father refused to accept that his child was not eligible for a place.

Ms Griffiths said: "He said he knew which my car was in the car park, he knew where I lived and that I had better offer the place or there would be consequences.

Give the head your support and give the teachers your support, any school will pull up and get better Madeline Brading, head teacher

"He would make sure that he got his revenge."

Over the past few years, the proportion of parents appealing successfully against a refusal of a place at a primary school has fallen.

In 1995-6, the figure was 48%. This dropped to 35% in 2000-1.

Of the head teachers questioned by the BBC, 72% said their workload had increased because of admissions-related work and greater demands from parents.

At Clare House Primary school in Bromley, Kent, Beverley Feather was offered �3,000 to ignore the fact that an address was outside the catchment area.

She added: "Parents will offer their skill and expertise. For instance, they say they could help with information and communication technology if their child got in.

"One time I was offered money: 'Would three thousand pounds help the decision-making process?' Which of course it wouldn't. It is solely based on distance from the home."

Dr Feather said she sympathised with parents, as choice of schools was really a "fiction".

Madeline Brading, head teacher of Our Lady of Victories Roman Catholic primary school in Kensington, west London, said the effort parents put into chasing places was largely misdirected.

She added: "If you go to a school and give the head your support and give the teachers your support, any school will pull up and get better."