The allegation was in an original version of his speech, which was leaked to the Daily Telegraph on Sunday, but was missing from the actual speech he delivered to the Institute for Public Policy Research (IPPR) in London yesterday.
Bryant's speech was coming on the back of Labour's criticism of the coalition's immigration policies. The party has criticised the use of vans bearing the slogan "go home" to try and encourage illegal immigrants to turn themselves in.
Bryant originally singled out the two retailers claiming they were "unscrupulous employers" only interested in finding cheap labour.
He accused Tesco and Next of recruiting large numbers of workers from low wage EU economies and bringing them to the UK.
He said the retailers charged the costs of their travel and their "substandard" accommodation against their wages and are 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.
Greater focus
In the actual speech he described Tesco as a "good employer and an important source of jobs in Britain".
He called for a greater focus on encouraging UK businesses to recruit locally.
"I want to see the government take action, working with companies to make sure they can recruit more local young people, [who are] qualified to do the job," Bryant said.
Bryant said UK employers had high levels of recruitment from abroad but provide far too low levels of training for local young people.
"Employers say they prefer to take on foreign workers. They have lots of 'get up and go', they are reliable, they turn up and they work hard," he said.
"But I've heard examples from across the country where employers appear to have made a deliberate decision not to provide training to local young people but to cut pay and conditions and to recruit from abroad instead, or to use tied accommodation and undercut the minimum wage," he said.
"So yes, we need British employers to do their bit - working to train and support local young people, avoiding agencies that only recruit from abroad, and shunning dodgy practices with accommodation or to get round the minimum wage."
The coalition said Labour was "badly confused" about immigration and lacked credibility.
In a statement, Tesco said it was pleased Bryant had recognised the firm as a good employer.
Next declined to comment.
