They were told job centres did not have the expertise to run training, so control of the budget should be transferred to the Learning and Skills Council, the quango which funds colleges and other training organisations for teenagers and adults.
The proposal was made in a paper which was presented to MPs by the Association of Learning Providers, which represents private training companies.
The ALP maintained that Jobcentre Plus should concentrate on finding jobs for the unemployed, leaving the LSC to help them acquire the skills to find work.
At the moment, the Department of Work and Pensions, which runs job centres, shares responsibility for training the unemployed with the LSC.
Graham Hoyle, the ALP's chief executive, said Jobcentre Plus had never developed the expertise for training since taking over work-based learning for adults in the 1990s. In the paper to MPs on the work and pensions select committee, he said: "It has never developed as effective a delivery infrastructure as the LSC.
"This is substantially because of the continued, cultural focus and priority on job finding as its key responsibility.
"Our proposal is that all funds for the support and training of unemployed people be transferred from the Department of Work and Pensions to the LSC.
"They use a sophisticated and ever more effective provider infrastructure, comprising both colleges and independent providers. This network already possesses the skills in all localities to fulfil this training role."
The problem has been exacerbated, the ALP added, by cuts in funding last year which affected training elements of New Deal and confused firms providing the training.
The ALP is one of several organisations called on to give evidence to the select committee about the effectiveness of Jobcentre Plus efficiency savings. Mr Hoyle added: "Acquiring basic skills in areas such as numeracy, literacy and IT is essential to help successful job-seekers to stay in work."