Mr Blair was delivering a speech to the National Association of Head Teachers, the first time a prime minister has addressed a teachers' union.
He said: 'The prime responsibility for improving schools lies with schools themselves - which is why LEAs should get as much money as possible out of central bureaucracy and into your schools. We aren't, frankly, satisfied that enough LEAs are doing this.'
He described the education authority spending figures, which are due for release this month, as 'league tables' and said: 'Next year David Blunkett will not hesitate to use his new powers to cap bureaucracy in LEAs which are unjustifiably holding back resources from schools.'
Education secretary David Blunkett reinforced the message in a letter to Local Government Association education chair Graham Lane.
'There remains deep concern, at school governor as well as at political level, that money which was hard to agree as part of the comprehensive spending review should in fact end up where the public thought it was going - into education,' Mr Blunkett said.
But he conceded that most councils have reflected standard spending assessment increases in their budget allocations.
Mr Lane said: 'Of nine authorities identified as underspending, eight of them have extremely good explanations. Norfolk is on the list and the government admit they were wrong about that.'
He said money was going to schools, it was just coming from other sources than the SSA because the government had ring-fenced much of the SSA. LGA figures on education authority spending would contradict the Department for Education and Employment's, he added.
'What's really going on is that the DfEE wants to spend money centrally.'