The claims were due to be made in a speech in London today by shadow immigration minister Chris Bryant (pictured), which has been leaked.
Tesco has asked the Labour party to remove the allegations.
According to the original speech, Bryant singled out the two retailers claiming they were "unscrupulous employers" only interested in finding cheap labour.
He accused Tesco and Next of recruiting workers in large numbers in low wage countries in the EU, bringing them to the UK, charging the costs of their travel and their "substandard" accommodation against their wages and not meeting the national minimum wage.
"That is unfair. It exploits migrant workers and it makes it impossible for settled workers with mortgages and a family to support at British prices to compete," Bryant was expected to say.
Bryant was going to use the example of Tesco's recent move to a new distribution centre in Kent, claiming most of the British staff at the original site were told they could only move if they took a pay cut.
This he claimed resulted in "a large" percentage of staff being recruited from Eastern Europe.
Polish workers
In further extracts from the speech Bryant was expected to say: "Look at Next, who last year brought 500 Polish workers to work in their South Elmsall warehouse for their summer sale and another 300 this summer.
"They were recruited in Poland and charged £50 to find them accommodation. The advantage to Next? They get to avoid Agency Workers Regulations which apply after a candidate has been employed for over 12 weeks, so Polish temps end up considerably cheaper than the local workforce which includes many former Next employees."
Bryant's speech later today is now expected to reflect discussions made with Tesco and Next.
Out of context
He has insisted that elements of it had been taken out of context.
He told the BBC's Today programme this morning he did not think the retailers were "unscrupulous" and was not going to engage in any kind of "xenophobia".
"They are good employers who often go the extra mile to try and include local workers," he said.
He said his criticism of employers who were effectively trafficking foreign workers, while pricing British employees out of the market, was not directed at Tesco and Next.
In a statement Next said: "Without access to the facts it is difficult to comment on what Bryant is claiming. On the face of it, his allegations seem unlikely."
Tesco called Bryant's claims "wrong" and said it "works incredibly hard to recruit from the local area".
