The Local Apprenticeships Service was initially planned as a pilot in a limited number of areas but would now be available to any English mayor who wants it, he said.
Through the scheme, which the Treasury says will roll out across England from March 2027, local teams plan to match young people with employers, including small businesses that have never taken on a trainee.
The department said an extra £100m was going into the scheme, “fully funded” by savings from the Department for Work and Pensions.
Powers to implement the scheme will rest with mayors because “they know which firms are hiring, which colleges are delivering, which estates have been left behind,” Healey said.
It comes as the final instalment of a major report by former minister Alan Milburn is expected to set out recommendations for tackling the one in eight people aged 16 to 24 in the UK who are Neets in the coming weeks.
Healey said these young people had been “signed off, written off” which the Labour government would address.
Helen Miller, director of the Institute for fiscal studies think tank, told the BBC that sticking to fiscal rules alone would not be enough to reduce debt because it is “basically a promise to get down debt tomorrow”.
She suggested that reducing the benefits bill, including removing support from some people, could be one way of cutting the national debt sooner.
Elsewhere in his speech, Healey repeated his promises to re-industrialise the country and said Rolls Royce would invest £300m more into its British factories in Derby, Bristol, Glasgow and Rotherham.
He also said he was bringing back the Union Learning Fund in England, established in 1998 and abolished by the Tories, to help working people gain more skills in an age of artificial intelligence. It was initially unclear how this would be funded.
