The approach has come from A4e, which is bidding for Home Office contracts for the Refugee Integration and Employment Service.
The three-year contracts in 12 regions across the UK are designed to provide help and support to refugees in establishing themselves in this country.
A4e, which already has government contracts finding jobs for unemployed people, told the refugee groups in an email that it would welcome their help in supporting service users. "What kind of support do you think you would be able to provide to us?" it asked.
A subsequent email made it clear there would be no money involved. "No payment will be made to your organisation; this is purely a referral partnership," it said.
Farrukh Husain, director of the Migrants Resource Centre, which was contacted by A4e, said the move was "gross exploitation of the voluntary sector".
He said: "A4e must think the voluntary sector is naive to simply wish to sign up to a relationship that delivers little in return for voluntary organisations and fattens up the profits of A4e."
An A4e spokeswoman said that if the organisation was subcontracting it would pay for services. But government guidelines about 'double funding' prevented the firm from paying for some services under the new contracts.
Husain responded that double funding was for front-line organisations to worry about. "It would be unethical for us to double count clients, and we do not," he said.
He said the Refugee Council, which was also bidding for the contracts, had made an approach to his organisation similar to that of A4e, but had offered payment.
Several other organisations, including the Haringey Somali Community and Cultural Organisation, have been approached by A4e with the same proposal made to the Migrants Resource Centre.
Maurice Wren, director of Asylum Aid - which is not involved - said he was concerned the affair could reduce the quality of bids.
"This could get some organisations thinking 'if we are to have any chance of getting a contract, we have to reduce our costs'," he said. "This means a poorer service for a very vulnerable client group."
Sara McKee, group commercial director at A4e, said: "The bid is still being developed, but we are actively looking for subcontractors within the third sector on the basis of a mutually beneficial partnership for all parties. A4e strongly believes that collaboration with the third sector is vital to reducing worklessness and supporting the hardest-to-reach in the UK."