Andrew Adonis wants Mr Blair to overrule sceptics in his own party and the Department for Education and Skills and give parents a voucher to spend at a school of their choice.
He believes vouchers would give parents a wider choice and help Labour outflank the Conservatives who announced their own voucher policy this week.
They could also help the Government fulfil its promise of a "personalised education" for every child - an idea flagged at last week's Labour party conference.
Mr Adonis, formerly head of the Prime Minister's policy unit and now his senior adviser on education and public services, is a staunch supporter of private involvement in public services.
He is understood to be disappointed that the Government has failed to attract more private-sector investment into education.
A source with close links to both the DfES and Downing Street told The TES that there was "constant traffic" between Number 10 and Milwaukee, home of America's oldest and largest voucher system.
He said the DfES was resisting the idea because it believed the practical problems outweighed the potential benefits.
Supporters of vouchers argue that they put power in the hands of parents by forcing schools to compete for their business.
But any attempt by Labour to introduce vouchers would be met by fierce opposition from backbenchers already angry over university tuition fees, foundation hospitals and the war on Iraq.
It would also be unpopular with many in education. The Conservatives' proposal was criticised by heads and teachers' leaders who argue that vouchers are bureaucratic, inefficient and place too much emphasis on private companies to provide schooling.
The National Union of Teachers has written to the Prime Minister asking him publicly to reject any form of school vouchers.
John Bangs, NUT head of education, said: "There will always be a temptation among Number 10 policy advisers to go for the market solution. So far the Prime Minister has resisted it. It would be a disaster if he went down the vouchers road."
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